metamaterials with

Metamaterials withNegative Parameters

Metamaterials withNegative ParametersTheory, Design, andMicrowave Applications

RICARDO MARQUES

FERRAN MARTIN

MARIO SOROLLA

Copyright # 2008 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Marques, Ricardo, 1954-.Metamaterials with negative parameters : theory, design and microwave applications / byRicardo Marques, Ferran Martin, Mario Sorolla.p. cm.

ISBN 978-0-471-74582-2 (cloth)1. Magnetic materials. 2. Microelectronics—Materials. I. Martin, Ferran,1965- II. Sorolla, Mario, 1958- III. Title.

TK7871.15.M3M37 2007620.1’1297—dc22

2007017343

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

http://www.copyright.com

http://www.wiley.com/go/permission

http://www.wiley.com

To our familiesAsuncion, Ricardo Jr., Concepcion and Ricardo Sr.

Anna, Alba and ArnauPuri, Carolina and Viviana

And also to the memory of Prof. Manuel Horno

Contents

Preface xiii

Acknowledgments xvii

1 The Electrodynamics of Left-Handed Media 1

1.1 Introduction 11.2 Wave Propagation in Left-Handed Media 21.3 Energy Density and Group Velocity 41.4 Negative Refraction 61.5 Fermat Principle 91.6 Other Effects in Left-Handed Media 9

1.6.1 Inverse Doppler Effect 101.6.2 Backward Cerenkov Radiation 101.6.3 Negative Goos–Hanchen Shift 12

1.7 Waves at Interfaces 13

1.7.1 Transmission and Reflection Coefficients 131.7.2 Surface Waves 15

1.8 Waves Through Left-Handed Slabs 16

1.8.1 Transmission and Reflection Coefficients 171.8.2 Guided Waves 171.8.3 Backward Leaky and Complex Waves 19

1.9 Slabs with 1/10!21 and m/m0!21 20

1.9.1 Phase Compensation and Amplification ofEvanescent Modes 20

1.9.2 Perfect Tunneling 211.9.3 The Perfect Lens 251.9.4 The Perfect Lens as a Tunneling/

Matching Device 29

1.10 Losses and Dispersion 32

vii

1.11 Indefinite Media 34Problems 35References 37

2 Synthesis of Bulk Metamaterials 43

2.1 Introduction 432.2 Scaling Plasmas at Microwave Frequencies 44

2.2.1 Metallic Waveguides and Plates as One- andTwo-Dimensional Plasmas 44

2.2.2 Wire Media 472.2.3 Spatial Dispersion in Wire Media 49

2.3 Synthesis of Negative Magnetic Permeability 51

2.3.1 Analysis of the Edge-Coupled SRR 522.3.2 Other SRR Designs 59

2.3.2.1 The Broadside-Coupled SRR 602.3.2.2 The Nonbianisotropic SRR 622.3.2.3 The Double-Split SRR 622.3.2.4 Spirals 62

2.3.3 Constitutive Relationships for Bulk SRR Metamaterials 652.3.4 Higher-Order Resonances in SRRs 702.3.5 Isotropic SRRs 732.3.6 Scaling Down SRRs to Infrared and Optical Frequencies 75

2.4 SRR-Based Left-Handed Metamaterials 80

2.4.1 One-Dimensional SRR-Based Left-HandedMetamaterials 81

2.4.2 Two-Dimensional and Three-Dimensional SRR-BasedLeft-Handed Metamaterials 85

2.4.3 On the Application of the Continuous-Medium Approach toDiscrete SRR-Based Left-Handed Metamaterials 87

2.4.4 The Superposition Hypothesis 882.4.5 On the Numerical Accuracy of the Developed Model for

SRR-Based Metamaterials 90

2.5 Other Approaches to Bulk Metamaterial Design 91

2.5.1 Ferrite Metamaterials 922.5.2 Chiral Metamaterials 972.5.3 Other Proposals 102

Appendix 107Problems 109References 114

3 Synthesis of Metamaterials in Planar Technology 119

3.1 Introduction 1193.2 The Dual (Backward) Transmission Line Concept 120

CONTENTSviii

3.3 Practical Implementation of Backward Transmission Lines 1283.4 Two-Dimensional (2D) Planar Metamaterials 1313.5 Design of Left-Handed Transmission Lines by Means of

SRRs: The Resonant Type Approach 135

3.5.1 Effective Negative Permeability Transmission Lines 1363.5.2 Left-Handed Transmission Lines in Microstrip and

CPW Technologies 1393.5.3 Size Reduction 144

3.6 Equivalent Circuit Models for SRRs Coupled toConventional Transmission Lines 146

3.6.1 Dispersion Diagrams 1513.6.2 Implications of the Model 151

3.7 Duality and Complementary Split Ring Resonators (CSRRs) 155

3.7.1 Electromagnetic Properties of CSRRs 1563.7.2 Numerical Calculation and Experimental Validation 160

3.8 Synthesis of Metamaterial Transmission Lines byUsing CSRRs 163

3.8.1 Negative Permittivity and Left-Handed Transmission Lines 1633.8.2 Equivalent Circuit Models for CSRR-Loaded

Transmission Lines 1663.8.3 Parameter Extraction 1703.8.4 Effects of Cell Geometry on Frequency Response 172

3.9 Comparison between the Circuit Models of Resonant-Type andDual Left-Handed Lines 175

Problems 180References 182

4 Microwave Applications of Metamaterial Concepts 187

4.1 Introduction 1874.2 Filters and Diplexers 188

4.2.1 Stopband Filters 1894.2.2 Planar Filters with Improved Stopband 1934.2.3 Narrow Bandpass Filter and Diplexer Design 198

4.2.3.1 Bandpass Filters Based on AlternateRight-/Left-Handed (ARLH) SectionsImplemented by Means of SRRs 199

4.2.3.2 Bandpass Filters and Diplexers Based onAlternate Right-/Left-Handed (ARLH) SectionsImplemented by Means of CSRRs 203

4.2.4 CSRR-Based Bandpass Filters with ControllableCharacteristics 207

4.2.4.1 Bandpass Filters Based on the Hybrid Approach:Design Methodology and Illustrative Examples 208

CONTENTS ix

4.2.4.2 Other CSRR-Based Filters Implemented byMeans of Right-Handed Sections 218

4.2.5 Highpass Filters and Ultrawide Bandpass Filters(UWBPFs) Implemented by Means of Resonant-TypeBalanced CRLH Metamaterial Transmission Lines 225

4.2.6 Tunable Filters Based on Varactor-Loaded SplitRings Resonators (VLSRRs) 227

4.2.6.1 Topology of the VLSRR and Equivalent-CircuitModel 228

4.2.6.2 Validation of the Model 2304.2.6.3 Some Illustrative Results: Tunable Notch

Filters and Stopband Filters 230

4.3 Synthesis of Metamaterial Transmission Lines with ControllableCharacteristics and Applications 233

4.3.1 Miniaturization of Microwave Components 2344.3.2 Compact Broadband Devices 2364.3.3 Dual-Band Components 2444.3.4 Coupled-Line Couplers 246

4.4 Antenna Applications 252Problems 258References 260

5 Advanced and Related Topics 267

5.1 Introduction 2675.2 SRR- and CSRR-Based Admittance Surfaces 268

5.2.1 Babinet Principle for a Single Split Ring Resonator 2685.2.2 Surface Admittance Approach for SRR Planar

Arrays 2705.2.3 Babinet Principle for CSRR Planar Arrays 2725.2.4 Behavior at Normal Incidence 2735.2.5 Behavior at General Incidence 274

5.3 Magneto- and Electro-Inductive Waves 278

5.3.1 The Magneto-Inductive Wave Equation 2795.3.2 Magneto-Inductive Surfaces 2825.3.3 Electro-Inductive Waves in CSRR Arrays 2845.3.4 Applications of Magneto- and Electro-Inductive Waves 285

5.4 Subdiffraction Imaging Devices 287

5.4.1 Some Universal Features of SubdiffractionImaging Devices 288

5.4.2 Imaging in the Quasielectrostatic Limit: Role ofSurface Plasmons 292

5.4.3 Imaging in the Quasimagnetostatic Limit: Role ofMagnetostatic Surface Waves 295

CONTENTSx

5.4.4 Imaging by Resonant Impedance Surfaces:Magneto-Inductive Lenses 299

5.4.5 Canalization Devices 302

Problems 304References 305

Index 309

CONTENTS xi

Preface

Discovery consists of seeing what everybody has seen and thinking what nobody hasthought.

Albert Szent-Gyorgyi

Classical electromagnetism is one of the best established theories of physics. Its con-cepts and theorems have been shown to be useful from the atomic to the cosmologicalscale; and they have been more successful in surviving to the relativistic and quantumrevolutions than other classical concepts. It is well known for instance that Maxwell’sequations were at the very basis of relativity and that Maxwell’s electromagnetictheory was the first relativistic invariant theory. Concerning quantum mechanics,classical electromagnetism still provides the best foundations—together withquantum dynamics—for atomic and solid state physics. It is only in the domain ofparticle physics that classical electromagnetism needs to be reformulated as quantumelectrodynamics. With regard to practical applications, classical electromagnetictheory is the basis of many well-known technologies, which strongly affect our ordin-ary lives, from power generation to wireless communications. It seems very difficult toadd something conceptually new to such well-established theories and technologies.

However, during recent years, a new expression appeared in the universe of clas-sical electromagnetic theory: metamaterials. From 2000 to 2007, the number ofjournal and conference papers related to metamaterials has grown exponentially;there has also been a multitude of special sessions, tutorials, and scientific meetings,all around the world, devoted to this new topic. Related to metamaterials, other topicsappeared on the scene, such as photonic crystals, negative refraction, left-handedmedia, or cloaking, among others. But what is the reason and meaning behind thissudden explosion in the otherwise quiet waters of electromagnetism? In fact,nothing is new from the point of view of fundamental science in metamaterials.Throughout this book, it will be shown that metamaterials can be understood byusing well-known theoretical tools, such as homogenization of effective media orelementary transmission line theories. In addition, many electrical and electronic

xiii

engineers have pointed out that almost all new applications arising from metamaterialconcepts can be understood by using more conventional approaches, that is, withoutthe need to invoke these metamaterial concepts. So, what is new in metamaterials?

Physicists usually try to explain how nature works, whereas engineers try to applythis knowledge to the design of new devices and systems, useful for certain appli-cations. In our opinion, metamaterials are placed at an intermediate positionbetween science and engineering—for this reason they are of interest to both physicistsand engineers. Metamaterials are not “materials” in the usual sense: they cannot befound in nature (by the way, this is a very common definition of metamaterials). Infact, metamaterials are artificial structures (products of human ingenuity), designedto obtain controllable electromagnetic or optical properties. This includes the possi-bility to synthesize artificial media with properties not found among natural materials,such as negative refraction, among others. Within this scenario, it is evident that meta-materials may open many challenging objectives of interest to physicists and scientistsin general. From the technological and engineering viewpoint, the interest in metama-terials is based on the possibility of designing devices and systems with new propertiesor functionalities, able to open up new fields of application or to improve existingones. Although it has been argued that certain applications of metamaterials can beanalyzed through conventional approaches, the key virtue of metamaterials is in pro-viding new design guidelines for components and systems that are missing in conven-tional approaches. Other applications such as subdiffraction imaging are, however,genuine products of metamaterials. This intermediate position between physics andengineering is a relevant aspect and probably one of the main novelties of meta-materials. In order to highlight this multidisciplinarity, in our opinion it is appropriateto refer to this new topic as metamaterials science and engineering.

Most metamaterials fall in one of two categories: photonic or electromagneticcrystals and effective media. The first category corresponds to structures made of per-iodic micro- or nano-inclusions whose period is of the same order as the signal wave-length. Therefore, their electromagnetic properties arise mainly from periodicity.Conversely, in effective media the period is much smaller than this signal wave-length. Hence, their electromagnetic properties can be obtained from a homogeni-zation procedure. This book is mainly devoted to the second category, specificallyto those metamaterials that can be characterized by a negative effective permittivityand/or permeability.

The first chapter is devoted to the analysis of the electrodynamics of continuousmedia with simultaneously negative dielectric permittivity and magnetic per-meability. Chapter 2 is focused on the design of bulk metamaterials made ofsystems of individual metallic inclusions with a strong electric and/or magneticresponse near its first resonance. The third chapter develops the transmission lineapproach for the design of metamaterials with negative parameters, including boththe nonresonant and the resonant-type approaches. Chapter 4 is devoted to theanalysis of some relevant microwave applications of the concepts developed in theprevious chapters. Finally, in Chapter 5, some related and/or advanced topics,such as metasurfaces, magneto-inductive waves in metamaterial structures and sub-diffraction imaging devices are developed.

xiv PREFACE

This book is mainly directed towards the resonant-type approach to metamaterialsbecause, obviously, it has been strongly influenced by the personal experience of theauthors. However, our aim in writing this book has been to give a complete overviewof the present state-of-the-art in metamaterials theory, as well as the most relevantmicrowave applications of metamaterial concepts. Indeed, our purpose has beentwofold: to generate curiosity and interest for this emerging field by those readersnot previously involved in metamaterials science and engineering and to provideuseful ideas and knowledge to scientists and engineers working in the field.

RICARDO MARQUES

FERRAN MARTIN

MARIO SOROLLA

Sevilla, SpainBarcelona, SpainPamplona, SpainOctober 2007

PREFACE xv

Acknowledgments

This book is the result of an intensive research activity in the field of metamaterials,which has been carried out by the authors and the members of their respective groups.We must acknowledge all of them because without their contribution this book wouldnever have been written.

With regard to the microwave group (Universidad de Sevilla), headed byFrancisco Medina, special thanks are given to Juan Baena and Manuel Freire, whohave significantly contributed to some parts of the book, including the editing ofmany figures. Also, special thanks are given to Francisco Medina, Francisco Mesa,Jesus Martel, and Lukas Jelinek for reading the manuscript and providing useful com-ments and suggestions. Finally, thanks to Rafael, Casti, Ana, Raul, Vicente, and allmembers of the group for providing such a friendly and stimulating environment forour research.

The members of CIMITEC (Center of Research on Metamaterials at theUniversitat Autonoma de Barcelona), headed by Ferran Martın, are also acknowl-edged, with special emphasis given to Jordi Bonache, Joan Garcıa, Ignacio Gil,Marta Gil, and Francisco Aznar, who have been actively and exhaustively involvedin this work. Special thanks are given to Marta Gil, who has helped the authorswith the generation and editing of some of the figures, and to Anna Cedenilla, forbeing in charge of copyright issues and permissions. We would also like tomention in the list the recently incorporated members to the team: Gerard, Fito,Ferran, and Beni.

Concerning the team of the Millimeter Wave Laboratory (Universidad Publica deNavarra), headed by Mario Sorolla, the authors thank Francisco Falcone, MiguelBeruete, Jose A. Marcotegui, Txema Lopetegi, Mikel A. G. Laso, Jesus Illescas,Israel Arnedo, Noelia Ortiz, Eduardo Jarauta, and Marıa Flores for their enthusiasticand creative research activities. Also, Mario Sorolla thanks Professor ManfredThumm (FZK and University of Karlsruhe) for his support and guidance overmany years in the topic of periodic structures.

The research activity presented in this book originated from several sources. At theEuropean level, thanks are given to the European Commission (VI Framework

xvii

Programme) for funding the Network of Excellence METAMORPHOSE, to whichone of the authors (Ferran Martın) belongs. Also, the authors have participated inthe European Eureka project, TELEMAC, devoted to the development ofmetamaterial-based microwave components for communication front-ends and sup-ported by the Spanish Ministry of Industry via PROFIT projects (headed by theSME CONATEL). At the national level, funding has been received from theMinistry of Science and Education (MEC) through several national projects.Special mention deserves the support given from MEC to the Spanish Network onMetamaterials (REME), which has been launched by the authors in collaborationwith other Spanish researchers.

And last, but not least, the authors would like to express their gratitude to theirrespective families for their understanding and support, and for accepting the largeamount of hours dedicated by the authors to the exciting field of metamaterialsand to writing the present manuscript.

R. M.F. M.M. S.

xviii ACKNOWLEDGMENTS

CHAPTER ONE

The Electrodynamics ofLeft-Handed Media

1.1 INTRODUCTION

Continuous media with negative parameters, that is, media with negative dielectricconstant, 1, or magnetic permeability, m, have long been known in electromagnetictheory. In fact, the Drude–Lorentz model (which is applicable to most materials) pre-dicts regions of negative 1 or m just above each resonance, provided losses are smallenough [1]. Although losses usually prevent the onset of this property in commondielectrics, media with negative 1 can be found in nature. The best-known examplesare low-loss plasmas, and metals and semiconductors at optical and infrared frequen-cies (sometimes called solid-state plasmas). Media with negative m are less commonin nature due to the weak magnetic interactions in most solid-state materials [2]. Onlyin ferrimagnetic materials are magnetic interactions strong enough (and losses smallenough) to produce regions of negative magnetic permeability. Ferrites magnetized tosaturation present a tensor magnetic permeability with negative elements near theferrimagnetic resonance (which usually occurs at microwave frequencies). Thesematerials are widely used in microwave engineering, mainly to make use of their non-reciprocity. The electrodynamics of these solid-state materials with negative 1 or m isdescribed in many well-known textbooks [3–5], so this chapter will focus on the elec-trodynamics of media having simultaneously negative 1 and m (see, however,problems 1.1–1.3, 1.12, and 1.14).

Wave propagation in media with simultaneously negative 1 and m was discussedand analyzed in a seminal paper by Veselago [6] at the end of the 1960s. However, itwas necessary to wait for more than 30 years to see the first practical realization of

Metamaterials with Negative Parameters. By Ricardo Marques, Ferran Martın, and Mario SorollaCopyright# 2008 John Wiley & Sons, Inc.

1

such media [7], also called left-handed media [6,7]. This terminology will beused throughout this book.1 Other terms that have been proposed for media withsimultaneously negative 1 and m are negative-refractive media [8], backwardmedia [9], double-negative media [10], and also Veselago media [11].

The propagation constant of a plane wave is given by k ¼ vffiffiffiffiffiffi1m

p, so it is apparent

that wave propagation is not forbidden in left-handed media. Several principal ques-tions arise from this statement:

† Does an electromagnetic wave in a left-handed medium differ in any essentialway from a wave in an ordinary medium with positive 1 and m?

† Is there any essential physical law—for instance, energy conservation—forbidding left-handed media?

† Assuming the answer to the previous question was affirmative. How to obtain aleft-handed medium in practice?

In this chapter we will try to give some answers to the first and second questionsabove, leaving the following chapters to find the answer to the third question.

1.2 WAVE PROPAGATION IN LEFT-HANDED MEDIA

In order to show wave propagation in left-handed media, we will first reduce Maxwellequations to the wave equation [6]:

r2 � n2

c2@2

@t2

� �c ¼ 0, (1:1)

where n is the refractive index, c is the velocity of light in vacuum, and n2=c2 ¼ 1m.As the squared refractive index n2 is not affected by a simultaneous change of sign in1 and m, it is clear that low-loss left-handed media must be transparent. In view of theabove equation, we can obtain the impression that solutions to equation (1.1) willremain unchanged after a simultaneous change of the signs of 1 and m. However,when Maxwell’s first-order differential equations are explicitly considered,

r� E ¼ �jvmH (1:2)

r�H ¼ jv1E, (1:3)

it becomes apparent that these solutions are quite different. In fact, for plane-wavefields of the kind E ¼ E0 exp(�jk � rþ jvt) and H ¼ H0 exp(�jk � rþ jvt) (thisspace and time field dependence will be implicitly assumed throughout this book),

1Terms such as right- and left-handed are of common use in optics, referring to polarized light or to mediahaving optical activity. Therefore, it will be worth noting that the use of the term left-handed in this book isnot related to any specific polarizing property or structural handedness of the material.

2 THE ELECTRODYNAMICS OF LEFT-HANDED MEDIA

the above equations reduce to

k � E ¼ vmH (1:4)

k �H ¼ �v1E: (1:5)

Therefore, for positive 1 and m, E, H, and k form a right-handed orthogonal systemof vectors. However, if 1 , 0 and m , 0, then equations (1.4) and (1.5) can berewritten as

k � E ¼ �vjmjH (1:6)

k �H ¼ vj1jE, (1:7)

showing that E, H, and k now form a left-handed triplet, as illustrated in Figure 1.1.In fact, this result is the original reason for the denomination of negative 1 and m

media as “left-handed” media [6].The main physical implication of the aforementioned analysis is backward-wave

propagation (for this reason, the term backward media has been also proposed formedia with negative 1 and m [9]). In fact, the direction of the time-averaged fluxof energy is determined by the real part of the Poynting vector,

S ¼ 12E�H�, (1:8)

which is unaffected by a simultaneous change of sign of 1 and m. Thus, E, H, and Sstill form a right-handed triplet in a left-handed medium. Therefore, in such media,energy and wavefronts travel in opposite directions (backward propagation).Backward-wave propagation is a well-known phenomenon that may appear in non-uniform waveguides [12,13]. However, backward-wave propagation in unboundedhomogeneous isotropic media seems to be a unique property of left-handed media.As will be shown, most of the surprising unique electromagnetic properties ofthese media arise from this backward propagation property.

FIGURE 1.1 Illustration of the system of vectors E, H, k, and S for a plane transverseelectromagnetic (TEM) wave in an ordinary (left) and a left-handed (right) medium.

1.2 WAVE PROPAGATION IN LEFT-HANDED MEDIA 3

So far, we have neglected losses. However, losses are unavoidable in any practicalmaterial. In the following, the effect of losses in plane-wave propagation will be con-sidered. To begin, we will consider a finite region filled by a homogeneous left-handed medium. In the steady state, and provided there are no sources inside theregion, there must be some power flow into the region in order to compensate forlosses. Thus, from the well-known complex Poynting theorem [14],

r � E�H�f g ¼ |v E � D� � B �H�ð Þ, (1:9)

it follows that

ReþE�H� � n dS

� �¼ v Im

ðmjHj2 � 1�jEj2

� �dV

� �, 0, (1:10)

where integration is carried out over the aforementioned region. Therefore,

Im(1) , 0; Im(m) , 0: (1:11)

Let us now consider a plane wave with square wavenumber k2 ¼ v2m1, propagatingin a lossy left-handed medium with Re(1) , 0 and Re(m) , 0. From expression(1.11), it follows that Im(k2) . 0. Therefore

{Re(k) . 0 and Im(k) . 0} or {Re(k) , 0 and Im(k) , 0}: (1:12)

That is, waves grow in the direction of propagation of the wavefronts. This fact is inagreement with the aforementioned backward-wave propagation.

1.3 ENERGY DENSITY AND GROUP VELOCITY

If negative values for 1 and m are introduced in the usual expression for the time-averaged density of energy in transparent nondispersive media, Und, given by

Und ¼14{1jEj2 þ mjHj2}, (1:13)

they produce the nonphysical result of a negative density of energy. However, as iswell known, any physical media other than vacuum must be dispersive [1], equation(1.13) being an approximation only valid for very weakly dispersive media. Thecorrect expression for a quasimonochromatic wavepacket traveling in a dispersive


media is [2]

U ¼ 14

@(v1)@v

jEj2 þ @(vm)@v

jHj2� �

, (1:14)

where the derivatives are evaluated at the central frequency of the wavepacket. Thus,the physical requirement of positive energy density implies that

@(v1)@v

. 0 and@(vm)@v

. 0, (1:15)

which are compatible with 1 , 0 and m , 0 provided @1=@v . j1j=v and@m=@v . jmj=v. Therefore, physical left-handed media must be highly dispersive.This fact is in agreement with the low-loss Drude–Lorentz model for 1 and m,which predicts negative values for 1 and/or m in the highly dispersive regions justabove the resonances [1]. Finally, it must be mentioned that the usual interpretationof the imaginary part of the complex Poynting theorem, which relates the flux of reac-tive power through a closed surface with the difference between electric and magneticenergies inside this surface [14], is not applicable to highly dispersive media, whereequation (1.14) holds instead of (1.13). Thus, this interpretation is also not valid forleft-handed media.

Backward-wave propagation implies opposite signs between phase and groupvelocities. In fact

@k2

@v¼ 2k

@k

@v; 2

v

vpvg, (1:16)

where vp ¼ v=k and vg ¼ @v=@k are the phase and group velocities, respectively. Inaddition, from k2 ¼ v21m and equation (1.15),

@k2

@v¼ v1

@(vm)@v

þ vm@(v1)@v

, 0 (1:17)

Finally, from equations (1.16) and (1.17)

vpvg , 0: (1:18)

This property implies that wavepackets and wavefronts travel in opposite directions,and can be considered an additional proof of backward-wave propagation in physicalleft-handed media.

1.3 ENERGY DENSITY AND GROUP VELOCITY 5

1.4 NEGATIVE REFRACTION

Let us now consider the refraction of an incident optical ray at the interface betweenordinary (1 . 0 and m . 0) and left-handed media. Boundary conditions imposecontinuity of the tangential components of the wavevector along the interface.Thus, from the aforesaid backward propagation in the left-handed region, it immedi-ately follows that, unlike in ordinary refraction, the angles of incidence and refractionmust have opposite signs. This effect is illustrated in Figure 1.2.

From the aforementioned continuity of the tangential components of the wave-vectors of the incident and refracted rays, it follows that (Fig. 1.2)

sin uisin ur

¼ �jk2jjk1j

;n2n1

, 0, (1:19)

which is the well-known Snell’s law. In this expression, n1 and n2 are the refractiveindices of the ordinary and left-handed media, respectively. Assuming n1 . 0, fromequation (1.19) it follows that n2 , 0. That is, the sign of the square root in the refrac-tive index definition must be chosen to be negative [6]:

n ; �cffiffiffiffiffiffi1m

p, 0: (1:20)

For this reason, left-handed media are also referred to as negative refractive index ornegative refractive media.

Geometrical optics of systems containing left-handed media is dominated by thislast property. By tracing the paths of optical rays through conventional lenses made ofleft-handed media, it can be shown that concave lenses become convergent and

FIGURE 1.2 Graphic demonstration of the negative refraction between ordinary (subscript1, top) and left-handed (subscript 2, gray area) media. Poynting and wavevectors for eachmedia are labeled as S1, S2, k1, and k2, respectively. Negative refraction arises from the conti-nuity of the components of the wavevectors, k1 and k2, parallel to the interface, and from thefact that rays propagate along the direction of energy flow. That is, they must be parallel to thePoynting vectors S1 and S2.


convex lenses become divergent, thus reversing the behavior of lenses made fromordinary media [6]. However, the most interesting effect that can be deduced fromthe geometrical optics of left-handed media is probably the focusing of energycoming from a point source by a left-handed slab [6]. This effect is illustrated inFigure 1.3. For paraxial rays

jnj ¼ j sin u1jj sin u2j

’ j tan u1jj tan u2j

¼ a0

a¼ b0

b, (1:21)

where n is the refractive index of the left-handed slab relative to the surroundingmedium. Therefore, as is illustrated in Figure 1.3, electromagnetic energy comingfrom the point source is focused at two points, one inside and the other outside theleft-handed slab, the latter at a distance

x ¼ aþ a0 þ b0 þ b ¼ d þ d

jnj (1:22)

from the source, where d is the width of the slab. If n ¼ 21 the aforementioned effectis not restricted to paraxial rays, because in this case ju1j ¼ ju2j for any angle of inci-dence. In fact, when n ¼ 21, all rays coming from the source are focused at twopoints, inside and outside the slab, the latter being at a distance 2d from thesource. We will return later to this interesting situation.

The above discussion mainly followed the earlier work of Veselago [6]. Morerecently, negative refraction at the interface between ordinary and left-handedmedia has been criticized on the basis of the highly dispersive nature of left-handed media [15]. In spite of these criticisms, negative refraction in left-handedmedia seems now to be well established. In fact, both theoretical calculations[16,17] and experiments [18–22] have confirmed Veselago’s predictions. Thereader interested in this specific topic can consult the aforementioned referencesfor a more complete discussion.

FIGURE 1.3 Graphic illustration of the focusing of paraxial rays coming from a point sourceby a left-handed slab. Light focuses at two points, F1 and F2, inside and outside the slab.

1.4 NEGATIVE REFRACTION 7

A brief discussion about the meaning and scope of the term negative refraction is ofinterest at this point. In the frame of this section, negative refraction and negativerefractive index are equivalent. That is, the incident and refracted beams always lieat the same side of the normal, and Snell’s law (1.19) is satisfied with a negative refrac-tive index that does not depend on the angle of incidence. In this strong sense, negativerefraction seems to be a unique property of isotropic left-handed media. In a lessrestrictive sense, negative refraction can also be understood as the simple situationwhere an incident and a refracted beam both lie at the same side of the normal, fora particular angle of incidence. However, it may be worth mentioning that, in thisweak sense, negative refraction can appear in many physical systems that may notinclude left-handed media, nor evenmedia with negative parameters. As an illustrationof this statement, Figure 1.4 shows the negative refraction of the extraordinary ray atthe interface between an ordinary isotropic medium and an hypothetical uniaxialmedium with 1? . 1k. The isofrequency ellipsoids, v(k) ¼ constant, for the isotropicand the uniaxial media are shown for a specific orientation of the optical axis. For thisspecific orientation, and for the considered specific angle of incidence, the group vel-ocity rk(v) corresponding to the wavevector of the refracted ray forms a negativeangle with the normal (in spite of the fact that the corresponding wavevector k r

makes a positive angle with the normal). Because the direction of the refracted rayis determined by the group velocity, the refracted ray forms a negative angle withthe normal. However, unlike negative refraction in left-handed media, negative refrac-tion in uniaxial media only occurs for very specific orientations of the optical axis, andfor very specific angles of incidence and polarizations of the incident light. A morerestrictive concept is all-angle negative refraction [23]. This means that a negativerefraction in the aforementioned weak sense is observed for all angles of incidence.All-angle negative refraction, with a refractive index depending on the angle of inci-dence, has been reported in uniaxial media with negative permittivity or permeabilityalong the optical axis [9] (see problem 1.8). Generalized refraction laws at the interface

FIGURE 1.4 Graphic demonstration of the negative refraction of the extraordinary ray at theinterface between an isotropic and an uniaxial medium. The isofrequency surfaces for bothmedia are shown, and the wavevector and the group velocity of the refracted ray are graphicallyobtained from these surfaces. The reported negative refraction only occurs for very specificangles of incidence and polarizations.


between anisotropic media with negative parameters are analyzed in detail in [24].Finally, it may be worth mentioning that negative refraction has been also reportedin strongly modulated photonic crystals for some specific frequency bands [25].

1.5 FERMAT PRINCIPLE

As is shown in many textbooks, Snell’s law can be deduced from the variationalFermat principle, and vice versa. The variational Fermat principle states that theoptical length of the path followed by light between two fixed points, A and B, isan extremum. The optical length is defined as the physical length multiplied by therefractive index of the material. Therefore, the Fermat principle is stated as

dL ¼ 0, where L ¼ðBA

n dl: (1:23)

Because Snell’s law (1.19) still holds for left-handed media, the Fermat principle willalso hold for systems containing left-handed media, provided the appropriate defi-nition for n (1.20) is taken in equation (1.23) [8]. It may be worth mentioning that,according to this statement of the Fermat principle, the optical length of the actualpath chosen by light is not necessarily a minimum when left-handed media arepresent. In fact, it may even be negative or null. It also becomes apparent that thetime taken by light to travel between two points is not related to the optical lengththrough the usual formula for ordinary weakly dispersive media, that is t = L=c.Therefore, the path followed by light in optical systems containing left-handedmedia is not necessarily the shortest in time [8].

From the definition of L (1.23), it follows that the optical length between two points,A and B, on a given optical ray is proportional to the phase advance between suchpoints: Df ¼ �kAB ¼ �n(v=c)AB ¼ �(v=c)L. It is of interest to evaluate theoptical length between the source and the focuses, F1 and F2, in the experimentshown in Figure 1.3 for the particular case of n ¼ 21. It follows immediately thatthe optical length between the source and the focus is exactly zero for all rays (notonly for paraxial rays). Thus, all the rays coming from the source are recovered at thefocus with exactly the same phase as at the source. Nevertheless, because the reflectioncoefficient for each ray depends on the angle of incidence, the intensity of the rays is notreproduced at the focus. However, if 1=10 ! �1 and m=m0 ! �1, the wave impe-dance of the slab becomes equal to that of free space, and the reflection coefficientvanishes for all rays. In such a case, it can be shown that the electromagnetic field atthe source is exactly reproduced at the focus [26]. We will return later to this point.

1.6 OTHER EFFECTS IN LEFT-HANDED MEDIA

Backward-wave propagation in left-handed media also has implications in other well-known physical effects related to electromagnetic wave propagation. The followingtext considers some of them.

1.6 OTHER EFFECTS IN LEFT-HANDED MEDIA 9

1.6.1 Inverse Doppler Effect

When a moving receiver detects the radiation coming from a source at rest in auniform medium, the detected frequency of the radiation depends on the relativevelocity of the emitter and the receiver. This is the well-known Doppler effect.If this relative velocity is much smaller than the velocity of light, a nonrelativisticanalysis suffices to describe such an effect. The qualitative analysis is quite simplefor ordinary (n. 0) media, and can be found in many textbooks. If the receivermoves towards the source, wavefronts and receiver move in opposite directions.Therefore, the frequency seen by the receiver will be higher than the frequencymeasured by an observer at rest. However, if the medium is a left-handed material,wave propagation is backward, and wavefronts move towards the source.Therefore, both the receiver and the wavefronts move in the same direction, andthe frequency measured at the receiver is smaller than the frequency measured byan observer at rest.

A straightforward calculation shows that the aforementioned frequency shifts aregiven by

Dv ¼ +v0v

vp, (1:24)

where v0 is the frequency of the radiation emitted by the source, v is the velocity atwhich the receiver moves towards the source, vp the phase velocity of light in themedium, and the + sign applies to ordinary/left-handed media. Equation (1.24)can be written in a more compact form as [6]

Dv ¼ v0nv

c, (1:25)

where n is the refractive index of the medium and c the velocity of light in free space.In equation (1.25), Dv is the difference between the frequency detected at the receiverand the frequency of oscillation of the source. For n, 0, the frequency shift becomesnegative for positive v (receiver moving towards the source), as it was qualitativelyshown at the beginning of this section. Interestingly, from equation (1.17), it directlyfollows that djkj=dv , 0 in left-handed media. Thus, a negative frequency shiftresults in an increase of jkj. Therefore, a shift towards shorter wavelengths is seenwhen the receiver approaches the source, both in ordinary and left-handed media.

1.6.2 Backward Cerenkov Radiation

Cerenkov radiation occurs when a charged particle enters an ordinary medium at avelocity higher than the velocity of light in such a medium. If the deceleration ofthis particle is not too high, its velocity can be considered approximately constantover many wave periods. Then, as is illustrated in Figure 1.5a, the spherical wave-fronts radiated by this particle become delayed with regard to the particle motion,


thus giving rise to a shock wave [1], which travels forward, making an angle u withthe particle velocity. This angle is given by

cos u ¼ c

nv, (1:26)

where c/n is the velocity of light in the medium and v the velocity of the particle.If the medium has a negative refractive index, wave propagation is backward, and

the spherical wavefronts corresponding to each frequency harmonic of the radiationmove inwards to the source, at a velocity c=jn(v)j. Therefore, each wavefrontcollapses at the advanced position of the particle shown in Figure 1.5b. Thus, theresulting shock wave travels backward at an obtuse angle from the particle motion.This angle is still given by equation (1.26) [6], as is illustrated in the figure.

In practice, any left-handed medium must be highly dispersive, the left-handedbehavior being restricted to some frequency range. Because the particle radiates atall frequencies, the Cerenkov radiation spectra must show wavefronts moving inboth forward and backward directions [27].

FIGURE 1.5 Illustration of the formation of Cerenkov shock waves: (a) in an ordinarymedium, and (b) in a left-handed medium. In (a) the spherical wavefronts move outwardsfrom the source at a velocity c/n. In (b) the spherical wavefronts move inwards to thesource at a velocity c/j nj.

1.6 OTHER EFFECTS IN LEFT-HANDED MEDIA 11

1.6.3 Negative Goos–Hanchen Shift

When a plane wave is incident from a medium of refractive index n ¼ n1 onto theplane interface of another medium with n ¼ n2, where jn2j , jn1j, there is a criticalangle sin uc ¼ jn2=n1j, beyond which total reflection onto medium 1 occurs.However, fields penetrate into medium 2 by a small distance, forming a nonuniformplane wave that is evanescent in the direction normal to the interface and propagativealong the interface. Power associated with this plane wave flows parallel to the inter-face in the forward direction for ordinary media and in the backward direction for left-handed media. Thus, when a beam of finite extent is incident from medium 1 tomedium 2, the reflected beam experiences a finite lateral shift D, as a consequenceof the aforementioned energy flow in medium 2. This effect is illustrated inFigure 1.6. As energy flow is parallel to wavefront propagation, the Goos–Hanchen shift must be positive in ordinary media. However, if medium 2 is a left-handed medium, energy flow and wavefront propagation are antiparallel.Therefore, the Goos–Hanchen shift must be negative in such media, as is illustratedin Figure 1.6b.

The Goos–Hanchen shift D can be calculated by expanding the incident beam inplane waves, and studying the reflection of these waves at the interface. If the angularspectrum of the beam is not too wide, and the angle of incidence is sufficiently awayfrom the critical and grazing angles, the Goos–Hanchen shift is given by [28]

D ¼ @fr

@kk, (1:27)

where fr is the phase of the reflection coefficient and k is the component of the wave-vector parallel to the interface (for lossless left-handed media jrj ¼ 1, andfr ¼ �j ln r). The Goos–Hanchen effect in left-handed media has been analyzedin detail in [29–32]. In [33], giant Goos–Hanchen shifts due to the excitation ofsurface waves have been reported for stratified media including left-handed layers.In all these calculations, the sign of the Goos–Hanchen shift agrees with the predic-tions of Figure 1.6.

FIGURE 1.6 Illustration of the Goos–Hanchen effect in (a) ordinary media and (b) left-handed media.


1.7 WAVES AT INTERFACES

So far, our analysis has been developed in the frame of the geometrical opticsapproximation, although with some general considerations regarding electromagneticwave analysis. In this section we will describe in more detail the electromagneticinteractions at the interface between a left-handed medium and an ordinarymedium. We will consider s-polarized waves, so that electromagnetic fields are trans-verse electric (or TE) along the normal to the interface (these waves are also calledlongitudinal section electric, or LSE, waves). The analysis of p-polarized (transversemagnetic, TM, or longitudinal section magnetic, LSM) waves is quite similar, and themain results can be deduced from the results for s-polarized waves by means of aduality transformation [14] (see Problem 1.7). In this and the following section wewill consider transparent lossless media. Thus, unless explicitly specified, losseswill be neglected, so that 1 and m will be real quantities. This approximation,although nonphysical, is useful when losses can be considered small and propagativeeffects dominate.

1.7.1 Transmission and Reflection Coefficients

Transmission and reflection coefficients at the interface between an ordinary mediumand a left-handed medium can be found using standard electromagnetic techniques.In this section we will follow the transverse transmission matrix [34] method. For thispurpose, plane waves at both sides of a plane interface will be decomposed intopropagative positive (þ) and negative (2) waves, with a common wavevectorcomponent parallel to the interface (the kz component in Fig. 1.7). For the plane

FIGURE 1.7 Definition of positive and negative waves for the determination of the trans-verse transmission matrix at the interface between an ordinary medium and a left-handedmedium. Note that backward propagation in the left-handed medium has been taken intoaccount in the definition of positive and negative waves.

1.7 WAVES AT INTERFACES 13

interface shown in Figure 1.7, and for the considered s-polarized waves,

Ei ; Ey,i ¼ Eþi þ E�

i , (1:28)

where subindex i ¼ 1 (i ¼ 2) stands for the fields at the left-(right-)hand side of theinterface (at x ¼ 0). Positive waves are defined as those waves carrying energy alongthe positive axis perpendicular to the interface. Thus, if the interface is perpendicularto the x-axis, as in Figure 1.7, the field dependence of positive and negative waves canbe summarized as

E+i ; E+

y,i / exp(+ jkx,ix� jkzzþ jvt), (1:29)

where kz is the common wavevector component parallel to the interface. According tobackward propagation, in the left-handed half space, kx,2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv212m2 � k2z

pmust be

chosen with Re(kx,2) , 0. For ordinary media kx,1 must be chosen with Re(kx,1) . 0,as usual.

A key magnitude in our analysis is the wave impedance of each medium, Zi, whichis defined as the ratio between the transverse (to x) components of the electric andmagnetic fields for positive waves propagating in such media. For the consideredTE waves, a straightforward calculation leads to the result [1,34]

Zi ;Eþi

Hþz,i

¼ vmi

kx,i, (1:30)

where Eþi ; Eþ

y,i is the electric field component of the positive wave. As both m2 andkx,2 are negative in the left-handed medium, the impedance (1.30) is positive, as isrequired for passive media.

After expanding the transverse electric and magnetic fields at the left (i ¼ 1)and the right (i ¼ 2) sides of the interface into positive and negative waves(Ei ¼ Eþ

i þ E�i and Hz,i ¼ Hþ

z,i þ H�z,i ¼ (Eþ

i � E�i )=Zi), and after imposing the con-

tinuity of the these field components at the interface, the transverse transmissionmatrix is determined as

Eþ1

E�1

� �; T

¼� Eþ

2E�2

� �¼ 1

2Z2

Z2 þ Z1 Z2 � Z1Z2 � Z1 Z2 þ Z1

� �� Eþ

2E�2

� �: (1:31)

From this result, the transmission, t, and reflection, r, coefficients are readily obtainedby taking Eþ

1 ¼ 1, E�1 ¼ r, Eþ

2 ¼ t, E�2 ¼ 0; that is,

1r

� �¼ 1

2Z2

Z2 þ Z1 Z2 � Z1Z2 � Z1 Z2 þ Z1

� �� t

0

� �, (1:32)


or

t ¼ 2Z2Z2 þ Z1

and r ¼ Z2 � Z1Z2 þ Z1

: (1:33)

For the particular case of Z2 ¼ Z1, from equations (1.33) it follows that t ¼ 1 andr ¼ 0. That is, the left-handed medium is perfectly matched with the ordinarymedium. From equations (1.30) it follows that this is the case for any angle ofincidence if 12=11 ! �1 and m2=m1 ! �1.

Equations (1.33) are also valid when losses in the left-handed half space aretaken into account. If, according to backward-wave propagation, we chooseRe(kx,2) , 0 in equation (1.29), from equation (1.12) it follows that Im(kx,2) , 0.Therefore, positive waves carry energy and attenuate along the positive x-axis, asis required from energy conservation. Substitution of these values of kx,2 inequations (1.30) and (1.33) gives the correct values for the transmission andreflection coefficients. Equations (1.33) are also valid for the total reflectioncase (i.e., when k2z . v212m2), provided kx,2 is chosen as kx,2 ¼ �ja,

a ¼ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2z � v212m2

p. In this case, Z2 (1.30) is imaginary, and t and r become

complex, with jrj ¼ 1, as is required from energy conservation.

1.7.2 Surface Waves

Surface waves along the interface between two different media must decay at bothsides of the interface. Therefore, imaginary values of kx,i should be considered.From the definitions of the previous section, it follows that

kx,i ¼ �jai and ai ¼ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2z � v2mi1i

q, (1:34)

in both ordinary and left-handed media. Therefore, positive waves in equation (1.29)will decay along the positive x-axis in both media. On the other hand, in order toobtain imaginary values of kx,i, it must be that

k2z . k21 , k22 , (1:35)

where k1 and k2 are the wavevectors in the unbounded ordinary and left-handedmedia, respectively (k2i ¼ v21imi). With these definitions, surface waves correspondto solutions of the following implicit equation (see also equation (1.31):

0E�1

� �¼ 1

2Z2

Z2 þ Z1 Z2 � Z1Z2 � Z1 Z2 þ Z1

� �� Eþ

20

� �: (1:36)

1.7 WAVES AT INTERFACES 15

This equation has nontrivial solutions if Z2 þ Z1 ¼ 0. According to equations (1.30)and (1.34), Z1 and Z2 are both imaginary and have opposite signs in ordinary and left-handed media. Therefore, solutions to equation (1.36) can exist at the interfacebetween these media, provided that expression (1.35) is satisfied. The generalcondition for the existence of such solutions is

k2z ¼m22 k

21 � m2

1 k22

m22 � m2

1

. k21 , k22 , (1:37)

which is also the dispersion relation for s-polarized (or TE, or LSE) surface waves. Asboth 12 and m2 strongly depend on frequency (see Section 1.3), the frequency depen-dence of kz(v) will depend on the specific characteristics of the left-handed medium.Dispersion curves for surface waves at the interface between free space andleft-handed media have been investigated in [35,36] for some realistic frequencydependences of 12 and m2. It may be worth mentioning that similar surface waves(sometimes called surface plasmons) are guided along the interface between ordinarymedia and media with negative 1 (see Problem 1.3). Surface waves can also be guidedalong the interfaces between ordinary and negative m media, and at the interfacebetween a medium with negative 1 and other with negative m [38].

Let us now consider again the limit 12=11 ! �1 and m2=m1 ! �1. In thislimit, the above dispersion relation becomes degenerate, being satisfied forany value of kz . k1 ¼ jk2j. Thus, in this limit, the bandwidth is zero and thedensity of states becomes infinite. This behavior is quite similar to that of surfaceplasmons at the surface of an ideal plasma in the quasistatic limit (see Problems1.2 and 1.3). However, unlike for surface plasmons, the degeneracy of the consideredsurface waves is complete, not being restricted to the quasistatic limit (i.e., tokz � k1 ¼ jk2j).

Finally, due to the particular form of the transverse transmission matrix inequations (1.31) and (1.36), mathematical solutions corresponding to nonphysicalsurface waves growing to infinity at both sides of the interface show the same dis-persion relation (1.37) as the aforementioned physical surface waves. Such solutions,although nonphysical for an isolated interface, will play an important role in theperfect lens configuration, as will be shown in the following.

1.8 WAVES THROUGH LEFT-HANDED SLABS

In this section, transmission and guidance of electromagnetic waves through left-handed slabs will be analyzed. The method of analysis will be a straightforwardextension of the transverse transmission matrix technique of the previous section.The same method can be applied to the analysis of transmission and guidance ofwaves through multilayered structures.


1.8.1 Transmission and Reflection Coefficients

The transmission matrix for a left-handed slab of width d is obtained by simply cas-cading the transmission matrices for each interface (1.31) and for the left-handedmedium in between,

Eþ1

E�1

� �¼ 1

4Z2Z3

Z2 þ Z1 Z2 � Z1Z2 � Z1 Z2 þ Z1

� �� e jkx,2d 0

0 e�jkx,2d

� �

�Z3 þ Z2 Z3 � Z2Z3 � Z2 Z3 þ Z2

� ��

Eþ3

E�3

� �, (1:38)

where, using the same notation as in equation (1.31), we have considered s-polarizedwaves. In equation (1.38) kx,2 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2m212 � k2z

p, 0 is the x-component of the

propagation constant for positive waves inside the slab, and subscripts 1, 2, and 3stand for the left-handed medium (2) and for the media at the left- and the right-hand sides of the slab (1 and 3). Zi are the wave impedances (1.30), and electricfields are evaluated just on the left (E+

1 ) and on the right (E+3 ) interfaces of the

slab. All waves inside and outside the slab have a common propagation vector, kz,parallel to the slab interface, which determines the angle of incidence sin ui ¼ kz=k1.

In the following, media 1 and 3 will be considered identical, so that Z3 ¼ Z1. Inthis case, the transmission and reflection coefficients, t and r, are obtained fromequation (1.38) by taking Eþ

1 ¼ 1, E�1 ¼ r, Eþ

3 ¼ t, E�3 ¼ 0. After a straightforward

calculation, the following expressions arise:

t ¼ 2Z1Z2j(Z2

1 þ Z22 ) sin(kx,2d)þ 2Z1Z2 cos(kx,2d)

(1:39)

and

r ¼ Z22 � Z2

1

Z22 þ Z2

1 � 2jZ1Z2 cot(kx,2d): (1:40)

Note that kx,2 , 0, so the phase advance through the slab is positive for small valuesof kx,2d, which corresponds to the propagation of a backward wave inside the slab.

1.8.2 Guided Waves

Waves guided along the slab correspond to the poles of the reflection coefficient(1.40) that may appear for imaginary values of kx,1. In such cases, according to ourprevious definitions, kx,1 ¼ �ja1, with a1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2z � v211m1

p. 0. Therefore,

1.8 WAVES THROUGH LEFT-HANDED SLABS 17

surface waves correspond to solutions of the implicit equation

Z22 þ Z2

1 � 2jZ1Z2 cot(kx,2d) ¼ 0, with Z1 ¼ jvm1

a1, (1:41)

with the additional condition kz . k1, so that electromagnetic energy decays at bothsides of the slab. Two cases may arise in equation (1.41), depending on whether kx,2 isreal or imaginary. If kz , k2 ¼ v

ffiffiffiffiffiffiffiffiffiffi12m2

p, kx,2 is real and fields inside the slab have a

trigonometric dependence, so that they are volume waves. In that case equation (1.41)reduces to

m2a1

m1 kx,2� m1 kx,2

m2a1þ 2 cot(kx,2d) ¼ 0; (1:42)

which, taking into account the trigonometric relation 2 cot(2x) ¼ cot x� tan x, can bewritten as

m2a1

m1 kx,2tan(kx,2d=2)þ 1

� �m1 kx,2m2a1

tan(kx,2d=2)� 1

� �¼ 0, (1:43)

which is the implicit dispersion relation for guided volume waves. In equation (1.43)the two branches, corresponding to symmetric and antisymmetric modes, are expli-citly shown. As can be seen from the above dispersion relations, volume waves inthe left-handed slabs do not substantially differ from guided volume waves in ordin-ary dielectric slabs [34]; waves inside the slab can be seen as the composition of twopropagating TEM waves, which suffer total reflection at the slab interfaces. Thereforeenergy flow is backward inside the left-handed slab and forward outside the slab, thewhole wave being forward or backward depending on the structural parameters of thewaveguide.

If kz . k2 ¼ vffiffiffiffiffiffiffiffiffiffi12m2

p, electromagnetic fields show an exponential decay inside

the slab, and the guided waves become surface waves. They correspond to the coup-ling between the surface waves generated at each slab interface (see Section 1.7.2).Therefore, when d ! 1, these waves should converge to the surface waves foundin Section 1.7.2. For these surface waves, both Z1 and Z2 are imaginary andcot(kx,2d) becomes a hyperbolic function. Thus, equation (1.41) reduces to

m2a1

m1a2þ m1a2

m2a1þ 2 coth(a2d) ¼ 0, (1:44)

which, taking into account the relation 2 coth(2x) ¼ coth xþ tanh x, can be written as

m2a1

m1a2tanh(a2d=2)þ 1

� �m1a2

m2a1tanh(a2d=2)þ 1

� �¼ 0, (1:45)


where the two branches, corresponding to symmetric and antisymmetric modes, canbe appreciated. The specific dependence of kz on frequency will depend on thespecific dependence of 1 and m on v. Surface and volume guided waves inleft-handed slabs, also including metallic ground shields, have been analyzed in[37–43]. Finally, it would be worth mentioning that resonant modes in other geome-tries distinct from planar ones, such as spherical or cylindrical, have been also studied(see, for instance, [44–47], and references therein).

1.8.3 Backward Leaky and Complex Waves

As in any open guiding system, most of the electromagnetic modes guided by a left-handed slab leak power to the surrounding space [13,34]. These waves correspondto the poles of the reflection coefficient (1.40) that may appear for complex valuesof kx,1.

Leaky modes in a left-handed slab have been analyzed in [48]. In the following wewill consider, without loss of generality, a leaky wave with field dependenceexp(�jkzzþ jvt), and Re(kz) . 0. The wavenumber of such a leaky mode shouldsatisfy Re(kz) , k1 ; v


p, so that power leaks at an angle cos u ¼ Re(kz)=k1

from the slab (Fig. 1.8). Inside the left-handed slab, power flow is backward, butin the surrounding medium, outside the slab, power flows in the forward direction.Therefore, for the analyzed leaky mode, power always leaks backward with regardto the main stream of guided power, inside the slab. For this reason, such leakymodes have been called backward leaky modes [48].

FIGURE 1.8 Illustration of the wavevectors and power flows in a backward leaky modeguided by a left-handed slab. Re(kz) is the mode phase vector, and Re(S) shows the powerflux inside and outside the slab. The angle of leakage is determined by cos u ¼ Re(kz)/k1.

1.8 WAVES THROUGH LEFT-HANDED SLABS 19

Physical leaky modes can be excited by sources at the onset of a semi-infiniteguiding system. In such a configuration, power is guided along the waveguide,from the source to its end. Considering Figure 1.8, this means that the source mustbe located at the top of the figure. Therefore, it must be that Im(kz) . 0, so thatthe leaky wave attenuates towards the lower end of the slab. On both sides of theslab the field dependence must be of the kind exp(+jkxx� kzzþ jvt). On theother hand, boundary conditions impose that k2z þ k2x ¼ k21, so that, for small attenu-ating constants,

k21 ’ {Re(kz)}2 þ {Re(kx)}

2 (1:46)

0 ¼ Re(kz)Im(kz)þ Re(kx)Im(kx): (1:47)

The first equation defines the leakage angle mentioned at the beginning of thissection. The second determines the sign of Im(kx) as follows: at the left-hand sideof the slab, power must flow to the left, so that Re(kz) . 0 and Re(kx) , 0.Moreover, Im(kz) . 0, as has already been mentioned. Therefore, it must be thatIm(kx) . 0, so that backward leaky waves are attenuated in the transverse direction.This result is somewhat unexpected, as leaky modes in ordinary guiding systems areusually unbounded modes, whose amplitude grows to infinity in the transverse direc-tion [34,49]. This unusual behavior of leaky modes in left-handed slabs is a directconsequence of the reported backward leakage of power [48], and recalls the behaviorof the complex modes that may appear in some very inhomogeneous wave-guidingsystems [13]. Backward leakage of power in a circuit analogous to an open left-handed waveguide has been experimentally reported in [50].

1.9 SLABS WITH 1/10!21 AND m/m0!21

As has already been mentioned, many striking properties of left-handed media arisein the limit when 1 and m become equal in magnitude, but opposite in sign, as inthose of the surrounding medium. In this section we will analyze these propertiesin detail. For simplicity, the surrounding medium is considered free space.

1.9.1 Phase Compensation and Amplification of Evanescent Modes

If 1=10 ! �1 and m=m0 ! �1, the left-handed wave impedances (1.30) of the left-handed medium become identical to that of free space for any propagative wave, andfor any angle of incidence. Therefore, for any angle of incidence, the transmissionmatrix at each slab interface (1.31) reduces to unity. Thus, the slab is perfectlymatched to free space. Even more interesting, because wave propagation is backwardinside the left-handed slab, the phase advance inside the slab is positive, and can beexactly compensated by the phase advance outside the slab (which must be negative).A straightforward calculation shows that this compensation actually happens—for


any angle of incidence—when the distance between the input and output planesis 2d, where d is the slab width. In fact, this last property is a direct consequenceof the zero optical length between the input and output planes already mentionedin Section 1.5

Let us now consider the incidence of evanescent waves on the slab. In this case,after substitution in equations (1.30), (1.39), and (1.40) (with kx,1 ¼ kx,2 ¼ �ja;a ¼ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2z � 10m0

p), it is readily shown that

t ¼ ead and r ¼ 0: (1:48)

Thus, evanescent plane waves are amplified inside the left-handed slab by a factor thatis just the attenuation constant at the input. Because evanescent modes do not carryenergy, this amplification, although striking, is not forbidden by energy conservation.Interestingly, when the input and output planes are separated by a distance 2d, theamplitude of any incident evanescent wave is recovered at the output. Thus, forthis specific location of the input and output planes, both propagating and evanescentwaves are recovered without change at the output.

Phase compensation and amplification of evanescent modes is not a unique prop-erty of left-handed slabs. After attention was first drawn to this effect by Pendry [26],it was also shown in a pair of consecutive slabs with only negative 1 and m [51,52]and, for some specific polarizations and planes of incidence, in some anisotropicslabs with negative parameters [53]. It has also been shown in a pair of coupledresonant surfaces [54]. Some of these results have been summarized in a generalizedperfect lens theorem [55].

1.9.2 Perfect Tunneling

As is well known, when a wave is incident from a medium of refractive index n ¼ n1onto the plane interface of another medium of n ¼ n2, where jn2j , jn1j, there is acritical angle uc ¼ sin�1jn2=n1j beyond which total reflection onto medium 1occurs. However, if medium 2 is substituted by a slab of finite thickness, and theslab is not too thick, some amount of energy can tunnel through medium 2 ontothe medium behind the slab. For ordinary media, the power tunneled through theslab is always smaller than the incident power, so there is also some reflectedpower. This tunneling of power is due to the coupling of evanescent waves generatedat both sides of the slab. Thus, because a left-handed slab can restore the amplitude ofevanescent waves (as was shown in the previous section), it can be guessed that tun-neling through left-handed slabs can be very significant. In fact, as will be shown inthis section, under some circumstances such tunneling of power can be so high that allthe incident power will flow through the slab. This “perfect tunneling” effect will beanalyzed in this section.

Instead of analyzing incidence in free space, a different but fully equivalent systemwill be analyzed. This system is shown in Figure 1.9, and consists of a rectangular wave-guide that includes a section filled with a left-handed medium with 1=10 ! �1 and

1.9 SLABS WITH 1/10!21 AND m/m0!21 21

m=m0 ! �1. As is well known [1,34], the fundamental mode in a rectangular wave-guide is the TE1,0 mode, which corresponds to the composition of two propagatings-polarized plane waves (E ¼ Ey y), which are reflected by the lateral sidewalls of thewaveguide, giving rise to a stationary wave in the transverse direction and a propagatingwave along the waveguide axis: Ey / sin(pz=a) exp(�jkxxþ jvt). This mode ispropagative/evanescent if thewaveguidewidth is larger/smaller than a half-wavelength(a_p=k, where k ¼ vj ffiffiffiffiffiffi

m1p j). Therefore, evanescent plane waves are generated in sec-

tions 2 to 4 of Figure 1.9 if a , p=k0, where k0 ¼ vffiffiffiffiffiffiffiffiffiffim010

p. In order to excite this eva-

nescent mode, we can simply place at the input a waveguide filled by a high-permittivitydielectric, supporting a propagative TE1,0 mode incident on section 2 (if the cross-sections of both waveguides are identical, no other modes will be generated at the inter-face [34]). It can be easily realized that such a system is equivalent to the incidence oftwo identical s-polarized waves coming from a semi-infinite medium (medium 1) onto aslab formed by media 2 to 4, and with a semi-infinite medium (medium 5) at the end ofthe system. The angle of incidence is u ¼ sin�1fp=(ka)g, where k2 ¼ v21m0, and inci-dence by an angle higher than the critical angle corresponds to the excitation of evanes-cent modes at waveguide sections 2 to 4 of Figure 1.9. Thus, a system such as that shownin this figure is fully equivalent, from an analysis standpoint, to a composite slabmade ofmedia 2 to 4, sandwiched between two semi-infinite media (1 and 5). However, thesystem of Figure 1.9 is easier to implement for an experimental verification oftunneling effects.

Let us now analyze the behavior of the device. When a propagative TE1,0 mode isincident from waveguide 1, evanescent TE1,0 modes are generated in waveguides 2 to4 and, eventually, some power may tunnel to waveguide 5. Following our previousanalysis, transmission and reflection coefficients can be obtained from the implicitequation

1

r

� �¼ 1

2Z2

Z2 þ Z1 Z2 � Z1Z2 � Z1 Z2 þ Z1

� �� ea2l2 0

0 e�a2l2

� �

� 12Z3

Z3 þ Z2 Z3 � Z2Z3 � Z2 Z3 þ Z2

� �� ea3l3 , 0

0, e�a3l3

� �

FIGURE 1.9 Illustration of a device designed to study perfect tunneling and amplification ofevanescent waves in a waveguide. The device consists of five consecutive waveguides with thesame cross-section. Input and output waveguides (1 and 5 ) are above cutoff. Waveguides 2 to 4are below cutoff. Waveguides 2 and 4 are empty. Waveguide 3 is filled with an isotropic left-handed medium with 1 ¼ �10, m ¼ �m0, and l3 ¼ l2 þ l4:


� 12Z4

Z2 þ Z3 Z2 � Z3Z2 � Z3 Z2 þ Z3

� �� ea2l4 0

0 e�a2l4

� �

� 12Z1

Z1 þ Z2 Z1 � Z2Z1 � Z2 Z1 þ Z2

� ��

t

0

� �, (1:49)

where ai are the attenuation constants of the TE1,0 mode in waveguides 2, 3, and 4

(ai ¼ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(p=a)2 � v2mi1i

q), and Zi are the wave impedances (1.30) for each

mode, with kx,i ¼ �jai,

Zi ¼ jvmi

ai; i ¼ 2, 3, 4: (1:50)

In the limit 1=10 ! �1 and m=m0 ! �1, Z3 ¼ �Z2, and we obtain

t ¼ 1 and r ¼ 0, for l2 þ l4 ¼ l3: (1:51)

That is, total transmission is obtained for the appropriate waveguide lengths. Thistotal transmission is not a Fabry–Perot resonance, because all waves between theinput and the output are evanescent. It is a perfect tunneling of power betweeninput and output [56].

Interestingly, the condition for perfect tunneling coincides with the condition forfocusing (1.22) when n¼21, and with the condition for amplitude and phase restor-ation (see Sections 1.4 and 1.9.1). In fact, if l2 þ l4 = l3, the amount of power tun-neled through the device decreases. This effect is illustrated in Figure 1.10, where the

FIGURE 1.10 Transmitted power through the device of Figure 1.9 versus the ratio(l2 þ l4)=l3 for several values of l3=l0 (values given close to the lines). The waveguidewidth is a ¼ 0:8l0=2. The dielectric constant of sections 1 and 5 is 1 ¼ 210. (Source:Reprinted with permission from [56]; copyright 2005 by the American Physical Society.)

1.9 SLABS WITH 1/10!21 AND m/m0!21 23

fraction of power tunneled through the device is shown as a function of (l2 þ l4)=l3.The sensitivity of the device to the normalized length of the left-handed slab l3=l0 (l0is the free-space wavelength at the frequency of operation) can also be observed. Ascan be expected, this sensitivity is higher for larger slabs.

In order to clarify the relation between perfect tunneling and amplification ofevanescent waves, it is illustrative to analyze field amplitudes along the device ofFigure 1.9 for the perfect tunneling configuration, and when the output waveguideis eliminated (l4 ! 1).2 Both amplitudes are shown in Figure 1.11. Amplificationof the evanescent wave inside the left-handed medium can be clearly appreciatedin Figure 1.11b. The restoration of the mode amplitude at a distance 2l3 from theinput can also be appreciated in this figure. However, the pattern for the wave ampli-tude is quite different in the perfect tunneling configuration (Fig. 1.11a). The reason

FIGURE 1.11 Field amplitude distribution inside the structure of Figure 1.9. (a) Perfect tun-neling when l3 ¼ l2 þ l4. (b) Field amplitude when the last transition is eliminated (l4 ! 1).Dashed lines show the amplitude when waveguide 3 is empty. Geometrical dimensions area ¼ 24mm, l2 ¼ l4 ¼ 15mm, l3 ¼ 30mm, and the operating frequency is 5GHz. The dielec-tric constant of sections 1 and 5 is 1 ¼ 210. (Source: Reprinted with permission from [56];copyright 2005 by the American Physical Society.)

2Such amplitudes can be obtained from the amplitudes at the input by cascading the appropriate trans-mission matrices between the input and the plane of interest.


for this is that single evanescent modes do not carry power, and perfect tunnelingimplies a maximum of power transmission. To obtain this transmission peak, thecomposition of two symmetrical evanescent modes decaying at opposite directionsis necessary. This leads to the amplitude pattern shown in Figure 1.11a.

Perfect tunneling of power through waveguides filled by metamaterials withnegative parameters was proposed and demonstrated in [56], where an experimentalverification of this effect was also provided. Photon tunneling through a left-handedslab in free space was analyzed in [57].

1.9.3 The Perfect Lens

The perfect lens is one of the most fascinating devices that can be designed by usingthe striking optical properties of left-handed materials. This configuration was firstproposed by Pendry in his seminal paper of 2000 [26]. In this paper Pendryshowed how the amplitude of evanescent waves can be restored by a left-handedslab, and how using this property, a perfect lens overcoming the diffraction limitfor the resolution of conventional lenses can be designed.

Before showing how the properties of left-handed slabs can be used to improvelens resolution, the limitations imposed by Fourier optics to the resolution of a lenswill be described in detail. Following [26], we will consider an infinitesimal dipoleof frequency v in front of a lens. The electric field produced by this dipole can beexpanded in some two-dimensional Fourier integral of plane waves (spatial Fourierharmonics),

Ep(x, y, z) ¼ð1�1

dky

ð1�1

dkz E(ky, kz) exp(�jkxx� jkyy� jkzzþ jvt), (1:52)

where kx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv210m0 � k2y � k2z

q. The response of any optical system to this excitation

can be obtained by analyzing the scattering of each plane wave of (1.52), and thenadding up for all the scattered waves. However, optical systems are usually only sen-sitive to propagating waves, that is, to waves with real values of kx (the x-axis isassumed to be the axis of the system). Evanescent waves are usually strongly attenu-ated and do not reach the image. Therefore, even for the best manufactured lens, theresolution at the image is limited by the truncation of the expansion (1.52) to

Ep(x, y, z) ’ð ðkmax

�kmax

dky dkz E(ky, kz) exp(�jkxx� jkyy� jkzzþ jvt), (1:53)

where kmax ¼ k0 ¼ vffiffiffiffiffiffiffiffiffiffi10m0

p. Equation (1.53) gives the effective source seen by the

optical system. Therefore, from a well-known rule of Fourier analysis, it immediatelyfollows that the aforementioned infinitesimal dipole is restored at the image with aresolution D not better than a wavelength (D � 2p=k0 ¼ l0).

The above analysis is based on the assumption that any lens or optical system willlose all the information contained in the evanescent Fourier harmonics of the source

1.9 SLABS WITH 1/10!21 AND m/m0!21 25

field expansion (1.52). However, we have already seen how a left-handed slab ofwidth d, and 1=10 ! �1, m=m0 ! �1 can restore the amplitude and phase ofany incident plane wave at a distance 2d from the input, both for propagative and eva-nescent waves. Therefore, it is the whole field at the source (1.52) that is restored atx ¼ 2d, not the truncated expansion (1.53). That is, left-handed slabs in the limit1=10 ! �1 and m=m0 ! �1 behave as perfect lenses, translating the whole field dis-tribution on a plane at the left side of the slab to another plane at the right side of theslab, as is shown in Figure 1.12.

It is illustrative to analyze the role of surface waves in the restoration of the eva-nescent Fourier harmonics at the image plane. Figure 1.13 shows the amplitudepattern for one of these evanescent Fourier harmonics between x ¼ 0 and x ¼ 2d.The physical and nonphysical surface waves mentioned at the end of Section 1.7.2can be easily recognized in the amplitude curves of Figure 1.13. Therefore, the res-toration of an evanescent Fourier harmonic in the perfect lens is due to the excitationof a physical surface wave at the right interface of the slab. This physical surface wavecouples with the evanescent fields generated at the source through the excitation of anonphysical (see the end of Section 1.7.2) surface wave3 at the left interface of theslab. It may be worth mentioning that this field pattern does not imply that anactual surface wave is generated at the slab. In fact, it can be easily realized thatequation (1.45) has no solutions for 1=10 ¼ m=m0 ¼ �1 (except for the trivialcase of d ! 1). In fact, the generation of such slab surface waves would be undesir-able, because it would imply a disproportionate contribution of the correspondingFourier harmonic to the image.

FIGURE 1.12 Illustration of the perfect lens proposed in [26]. The fields at x ¼ 0 (E(0, y, z)and H(0, y, z)) are exactly reproduced at x ¼ 2d: E(0, y, z) ¼ E(2d, y, z) and H(0, y, z) ¼H(2d, y, z), provided there are no sources between the planes x ¼ 0 and x ¼ 2d.

3The denomination “nonphysical” indicates that these waves do not satisfy physical boundary conditions atinfinity when they are considered in an unbounded system (they monotonically grow at both sides of theinterface). However, the presence of additional boundary conditions, as in Figure 1.13, allows for itsexcitation.


Some considerations should be made at this point. First of all, as has already beenmentioned, restoration of evanescent Fourier harmonics at x ¼ 2d implies the ampli-fication of such Fourier harmonics inside the slab (Fig. 1.13). As we approach theback interface of the slab, the density of energy stored by these evanescent wavesgrows by the exponential factor exp[2a(d � d)], where d is the distance to the inter-

face. In the limit jkzj, jkyj ! 1, we have a(ky, kz) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y � v210m0

q! 1, and

the density of energy grows to infinity at the slab interface. Therefore, the perfect lensimplies a nonphysical divergence of energy [58]. Although we will return to thispoint later (see Section 1.10) it should be mentioned here that this apparentparadox disappears if lossy slabs are considered [59–61]. An additional effect oflosses is to limit the resolution of the perfect lens. Lossy left-handed slab lensescan still improve the value of kmax (kmax . v


p) in equation (1.53), but

leaving it finite (see Section 1.10 for a complete discussion on this point).Therefore, it will be more appropriate to speak about lossy left-handed superlenses,which approach Pendry’s perfect lens when losses approach to zero.

Other considerations of interest arise from the comparison between the Veselagolens (see Fig. 1.3) and Pendry’s perfect lens. For this purpose we will consider theimage of a point source created by both lenses. A Veselago lens, with n , 0 but1=10 ,m=m0 = �1, will be considered first. It is a nonconventional geometricaloptics system, where electromagnetic energy coming from a point source isfocused into a three-dimensional spot beyond the lens, of radius not smaller than ahalf wavelenght (this behavior is sketched in Fig. 1.14a). Therefore, superresolutionis not present in the Veselago lens. Let us now consider Pendry’s perfect lens. Inthis device, the fields at x ¼ 0, where the source is placed, are exactly reproducedat x ¼ 2d (see Fig. 1.12). Therefore, from the uniqueness theorem [14], it immedi-ately follows that the source fields in the x-interval [0, 1] are exactly reproducedbeyond the lens in the x-interval [2d, 1]. However, nothing ensures that the source

FIGURE 1.13 Amplitude pattern for an evanescent Fourier harmonic of the perfect lensshown in Figure 1.12. The amplitude follows the curves exp(+ax), where

a(ky, kz) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2y þ k2z � v210m0

q.

1.9 SLABS WITH 1/10!21 AND m/m0!21 27

fields for x , 0 will be reproduced at x, 2d. In fact, this will be impossible, as thepresence of the source implies a field divergence at x ¼ 0 that is not present at x ¼ 2d[62]. Intuitively, it can be guessed from the amplitude pattern of Figure 1.13 thatfields must grow from the image plane to the lens, regardless of behavior near thesource (such behavior is sketched in Fig. 1.14b). In fact, this behavior has been con-firmed by analysis [63] and experiment [64]. Therefore, Pendry’s perfect lens doesnot focus the energy coming from a point source into any spot in three-dimensionalspace. Nevertheless, it still produces a two-dimensional spot on the image plane (x ¼2d ), of a size that can be much smaller than a square wavelength. Therefore, Pendry’sperfect lens produces two-dimensional superresolution in the image plane, but notsuperfocusing of energy in three-dimensional space4. This characteristic of theperfect lens (and similar devices) imposes severe limitations to its applicability forsubsurface heating or burning devices. It also imposes restrictions on its usefulness

FIGURE 1.14 Illustration of the behavior of the Veselago (a) and Pendry (b) lenses. In theVeselago lens the energy coming from a point source, S, is focused onto a spot at the focus, F,at a distance d þ d=jnj from the source. The size of this spot is not smaller than a wavelength.In Pendry’s perfect lens, a virtual image, S0, of a point source, S, is created at a distance 2d fromS. However, fields at x , 2d grow exponentially towards the lens, so there is no focusing ofenergy at any point.

4Such three-dimensional superfocusing will be contrary to the well-known uncertainty principle, as hasalready been noted in [65] and [66]. It should also be mentioned that any practical lens must be lossy,and that the behavior of lossy Pendry’s superlenses approaches that of Veselago lenses when the ratiod/l0 increases (see Section 1.10).


in imaging devices, at least if the distance between the source and the lens isunknown. In fact, because the electromagnetic field intensity has a monotonicdecay at the back side of the lens, the location of an unknown source can hardlybe recovered from the analysis of the field distribution at the back side of the lens(see [63] for a complete discussion, with examples, of this point).

Experimental confirmation of subdiffraction imaging in Pendry’s perfect lens hasbeen reported in [64], and the effects of an input or output medium other than freespace have been investigated in [67] (see Problem 1.13). Superresolution similar tothat produced by a left-handed perfect lens, but in the quasistatic limit, can beobtained by using slabs with only negative 1 (such as silver plates at optical frequen-cies) [26,68–71] (see Problem 1.14). It can be also obtained in magnetized ferriteslabs [72] and in pair-coupled magnetoinductive surfaces [73].

1.9.4 The Perfect Lens as a Tunneling/Matching Device

In the previous section, the field created by a source placed in front of a perfect lenswas studied. In this section we will address the problem of image detection in suchdevices. A key point in the behavior of Pendry’s perfect lens is that most of the infor-mation reaching the image plane is carried by evanescent electromagnetic waves (theaforementioned evanescent Fourier harmonics). As is well known, a single evanes-cent electromagnetic wave does not carry power (for this reason the amplificationof evanescent Fourier harmonics inside the lens does not violate energy conserva-tion). However, any measurement of the image should imply the transmission ofsome amount of power from the source to the detector. Power transmission by eva-nescent waves is known in physics as tunneling. Therefore, seen from the detector,a perfect lens is a kind of tunneling device. For a single electromagnetic mode, tun-neling of power was analyzed in Section 1.9.2. For a summation of evanescent waves(the aforementioned evanescent Fourier harmonics), the basic mechanism should beessentially the same: Some new evanescent waves are generated at the output inaddition to the evanescent Fourier harmonics generated by the source. It is the com-position of all these waves that allows for power transmission. The aforementionednew evanescent waves generated at the output are not present before the detector isplaced at the output. Therefore, the image detection process substantially affectsthe field distribution along the system formed by the source and the lens. Thesekind of devices, where fields depends on both the input and the output, have longbeen considered by electrical engineers. Some of these devices—those that maximizepower flow—are usually called matching devices. Thus, from this standpoint, theperfect lens can also be considered a matching device [66,74].

In the following we will analyze the image detection process in detail. As usual,we will consider a system formed by the perfect lens, an input antenna (the source S ),and an output antenna (the detector D). Two waveguides are connected to the inputand output antennas (for simplicity both waveguides will be assumed identical, withcharacteristic impedance Z0). The image is obtained by scanning the detector behindthe lens, as a map of the transmission coefficient between the source and the detector.The whole system is illustrated in Figure 1.15. The computation of the transmission

1.9 SLABS WITH 1/10!21 AND m/m0!21 29

coefficient is a standard problem of electromagnetic theory, which can be solved ifthe impedance matrix Zij of the system formed by the antennas and the lens isknown. The resulting expression is [75]

t ¼ 2Z12Z0(Z11 þ Z0)(Z22 þ Z0)� Z2

12

, (1:54)

where Zij are the elements of the impedance matrix of the system formed by the twoantennas and the left-handed slab, and Z0 is the characteristic impedance of the input/output waveguides. If superresolution has to be detected, the size of the input andoutput antennas has to be smaller than the free-space wavelength, which wouldalso make the real part of Zij (the radiation resistance) negligible with respect to itsimaginary part [14]; in other words, Zij ’ jXij, where Xij is the reactance matrix ofthe system5. Moreover, because the perfect lens perfectly matches any source tofree space, X11 and X22 can be approximated by the reactances of the input andoutput antennas in free space. The off-diagonal term, X12, accounts for the mutualreactance between the antennas, which is strongly affected by the presence of thelens. Therefore,

t � 2jX12Z0( jX11 þ Z0)( jX22 þ Z0)þ X2

12

: (1:55)

If one of the self-reactances dominates over the remaining ones, the first term inthe denominator of (1.55) dominates over the second, and

t / X12: (1:56)

FIGURE 1.15 Experimental setup for the detection of images created by a left-handedperfect lens. The source S is placed at a fixed distance from the left-handed slab, and the detec-tor D is scanned behind the slab. The transmission coefficient between S and D is measured.

5Throughout this section we will consider lossless antennas. Loading the antennas with a resistance hasconsequences that will not be analyzed here. See [63] for an analysis of this particular scenario.


That is, the transmission coefficient is simply proportional to the mutual reactance,X12, which usually provides a good measure of the field in the absence of theoutput. This case may correspond to a small electric dipole antenna at the output(in such a case, jX22j � jX21j, jX11j). It may also occur when the output antenna isa small magnetic loop (in such a case, usually, jX11j � jX21j, jX11j, [63]).Therefore, if small electric or magnetic probes are used as detectors, the transmissioncoefficent through the device can give a correct measurement of the field created bythe source at the back side of the lens. This result can be expected from the physics ofthe experiment; in both cases the field generated by the currents at the output antennashould be very small, and should not substantially affect either the current at the inputantenna or the field distribution behind the lens. As was shown in [64], when such asmall probe is used for field measurements at the back side of the lens, the maximumof transmission is obtained just at the back interface of the lens, in agreement with thebehavior outlined in Figure 1.14.

However, if the self-reactances of the input and output antennas are the same(X11 ¼ X22 ¼ X ), which may corresponds to a detector identical to the source, then

t � 2jX12Z0X212 � X2 þ Z2

0 þ 2jXZ0: (1:57)

The values of t, as a function of X12 and X, are plotted in Figure 1.16. Except for verysmall values of X, the maximum is achieved for X12 � X, and in such a case it is alsothe case that jtj � 1. Because both antennas are identical and the lens reproduces thefield of the source at the image plane, the condition X12 � X is fulfilled when thedetector is placed at the theoretical location of the image, at a distance 2d from thesource. Therefore, if the source and the detector are identical, and the detector is

FIGURE 1.16 Plot of the modulus of the transmission coefficient t ¼ S21 (1.57) as a func-tion of the modulus of the reactances X11 ¼ X22 ¼ X and X21 (Source: Reprinted with per-mission from [63]; copyright 2005 by the American Physical Society.)

1.9 SLABS WITH 1/10!21 AND m/m0!21 31

placed at the theoretical location of the image (i.e., at a distance 2d from the source),the perfect lens behaves as a perfect tunneling (or perfect matching) device, giving amaximum of power transmission at the theoretical location of the image (i.e., at thepoint marked as S0 in Fig. 1.14). If the antennas are not identical, but none can beconsidered a small probe (in the sense defined in the previous paragraph), the locationand magnitude of the maximum transmission will vary. This tunneling effect in meta-material perfect lenses and similar devices has been analyzed in detail in [63]. Theexperimental confirmation of such an effect has been shown in a device similar toPendry’s super-lens [73] (see also Chapter 5).

Two main conclusions arise from the above analysis. The first is that for fieldmeasurements at the back side of a metamaterial super-lens, care must be taken inorder to avoid artifacts coming from the matching/tunneling capabilities of suchdevices. The second is that, in some circumstances, it may be possible to take advan-tage of such a tunneling effect in order to obtain additional information from theimage. For instance, if the shape and characteristics of the source are known, it ispossible to design an appropriate detector in order to determine the source locationin three-dimensional space from measurements of the transmission coefficientbehind the lens.

1.10 LOSSES AND DISPERSION

After the first seminal papers showing propagation in left-handed media, it wasclaimed that losses and dispersion will destroy many of the previously reportedeffects. Specifically, negative refraction [15] and super-resolution in Pendry’sperfect lens [58] were criticized. The analysis of the refraction of a Gaussian beamat the interface between an ordinary and a left-handed medium showed, withoutdoubt, that negative refraction occurs in such a situation [16,17], thus confirmingthe Veselago analysis [6]. Of more interest are the effects of losses on the perfectlens proposed by Pendry [26]. This effect was analyzed in [59–61] among others,leading to similar conclusions. In the following we will briefly examine this aspect.

In order to develop the analysis, it is convenient to rewrite the transmission coeffi-cient (1.39) as

t ¼ 4Z

(1þ Z)2 exp( jkx,2d)� (1� Z)2 exp(�jkx,2d), (1:58)

where Z ¼ Z2=Z1. If 1=10 ! �1 and m=m0 ! �1, from our definitions of positivewaves6 and from equation (1.30), it follows that Z ¼ 1 if kx,2 ¼ �jkx,2j is real, andZ ¼ �1 if kx,2 ¼ �ja is imaginary. Therefore t ! exp( jjkx,2jd) for propagativewaves and t ! exp(ad) for evanescent waves. In both cases, the phase and amplitude

6According to our definitions (see Fig. 1.7 as well as Section 1.7 and equation 1.12), in the left-handed slab,propagation is backward and positive waves are defined with Re(kx,2), 0 and/or Im(kx,2) , 0. In ordinarymedia, positive waves are defined with Re(kx,2) . 0 and/or Im(kx,2) , 0, as usual.


of the incident waves change along the slab just by the amount necessary to produceFourier harmonics restoration at a distance 2d from any source (see Section 1.9.3).Let us now introduce a small amount of losses in the left-handed slab, so that1 ! �10(1þ jd1) and m ! �m0(1þ jdm). Let us also suppose that kz � k0, sothat kx,2 ¼ �ja ’ �jjkzj. In such limit, taking into account equation (1.30), we obtain

t ! 4

4 exp(�jkzjd)� d2m exp(jkzjd): (1:59)

Therefore, for high values of kz it is still t ’ exp(jkzjd) ’ exp(ad) providedd2m exp(jkzjd) , 4 exp(�jkzjd), or

jkzjd . ln2dm

� �: (1:60)

However, if the above condition is not fulfilled, then t � exp(�jkzjd), and evanescentwaves are not amplified inside the left-handed slab. A similar (dual) result holds forp-polarized (or TM) modes.

Regarding the analysis of Section 1.9.3, the above results imply that, in general, alossy left-handed slab with 1=10 ! �(1þ jd1) and m=m0 ! �(1þ jdm) will see theeffective source (1.53) with

kmax �1dln

2d

� �; d ¼ max d1, dm

� , (1:61)

so that the source is seen at the image with a resolution D given by

D

d&

2pkmaxd

¼ 2p ln2d

� �� 1

: (1:62)

It is worth noting that the wavelength of the incident radiation does notappear in equation (1.62), only the losses and thickness limit the resolution of thedevice.

An interesting consequence of the above results is that losses prevent the diver-gence of energy mentioned in section 1.9.3. In fact, because Fourier harmonics areonly restored upto some finite value of jkzj ¼ kmax, the energy divergence associatedwith the harmonics in the limit jkzj ! 1 disappears. However, this regularizationtakes place at the cost of a loss in resolution. Thus, the lossy perfect lens is, infact, a super-lens, with a resolution limited by equation (1.62). As the free-spacewavelength is not present in equation (1.62), the resolution of such a lens may stillovercome the diffraction limit (D . l0). However, the logarithmic dependence on1=d of D=d strongly limits this effect. In fact, even in the best dielectrics, d1 is not

1.10 LOSSES AND DISPERSION 33

much smaller than 1023. Therefore, as a consequence of equation (1.62), super-resolution can only be present in left-handed slabs with a width of only a fractionof the wavelength. Thus, in practice, any left-handed super-lens is a near-fielddevice whose effectiveness should be measured with regard to other near-fieldmicroscopy devices [76]. In order to reduce the limitations caused by losses in thesuper-lens performance, multilayer lenses formed by alternating layers of ordinaryand left-handed media have been proposed [77,78]. Such devices provide a largerdistance between the source and the image, but not a larger distance between thesource and the left interface of the lens.

The above analysis can easily be extended to variations in the real part of 1 and m:1 ! �10(1þ d1) and m ! �m0(1þ dm) [59]. The results are quite similar to thosereported in the previous paragraph. The main difference is that these small variationsof 1 or m may allow for the excitation of slab guided waves (see Section 1.8.2) forsome values of the transverse wavenumber. In such cases, very high values of thetransmission coefficient are obtained for the corresponding spatial Fourier harmonic[59], which may distort the image. As left-handed metamaterials are highly dispersivemedia, this effect may appear for sources emitting radiation pulses of finite duration.

The effect of the finite duration of RF pulses in the focusing of rays in theVeselago lens was also analyzed in [61]. The main conclusion is that, for the effectivefocusing of a ray, the pulse duration should be longer than the time taken by light totravel between the source and the focus along such a ray.

1.11 INDEFINITE MEDIA

Upto now, we have focused our attention on isotropic left-handed media. However,most present realizations of such media are in fact highly anisotropic, showing aleft-handed behavior only for very specific polarizations and directions of propa-gation. These structures (some of them will be described in detail in the followingchapters of this book) are easier to manufacture than isotropic left-handed media,while keeping many of their interesting properties. Therefore, they are of great prac-tical interest. Many of them can be described by means of symmetric and simul-taneously diagonalizable permittivity, 1

¼, and permeability, m

¼, tensors with some

negative eigenvalues. These media have been called indefinite media [53], a denomi-nation that emphasizes that not all the eigenvalues of the constitutive tensors have thesame sign. In this section, wave propagation in indefinite media will be briefly dis-cussed. This discussion will mainly follow the analysis in [53].

In order to simplify the analysis, we will choose a system of reference such thatboth 1

¼and m

¼are diagonal:

1¼ ¼

1x 0 00 1y 00 0 1z

0@

1A; m

¼ ¼mx 0 00 my 00 0 mz

0@

1A: (1:63)


Because the dispersion relation for plane waves of generic polarization propagating insuch a medium may be rather complicated, we will focus our attention onsome particular cases of interest. In particular we will consider the very importantcase of TE waves polarized along a main axis of 1

¼and m

¼. As in the previous sections

we will choose, without loss of generality, E ¼ E y. The dispersion relation for suchwaves is [53]

v2 ¼k2z

1ymxþ k2x1ymz

(1:64)

(the dispersion relation for TM waves is the dual one). Taking v as a parameter,equation (1.64) defines the isofrequency curves in the (kz, kx) plane. These curvesmay be ellipses or hyperbolae, depending on the signs of 1ymx and 1ymz. It may beworth mentioning that, because left-handed media are highly dispersive, the consti-tutive parameters in equations (1.63) and (1.64) are usually frequency dependent.Therefore, in practical indefinite media, the shape of these curves may changefrom hyperbolic to elliptic, and vice versa, depending on frequency.

The direction of propagation of optical rays and energy is defined by the groupvelocity vg ¼ rkv(k). From equation (1.64) it is readily found that

vg ¼1v

kx1ymz

, 0,kz

1ymx

� �: (1:65)

The angle between group and phase (vp ¼ kv=k2) velocities is defined by

vp � vg ¼1k2

k2x1ymz

þk2z

1ymx

� �, (1:66)

which generalizes equation (1.18). The laws for the refraction of optical rays at theinterface of an indefinite media can be deduced from equation (1.64) by imposingthe continuity of the tangential component of the wavevector k at the interface,and using equation (1.66) to draw the rays in the indefinite media [24] (seeFig. 1.4 and Problem 1.8). Equation (1.64) is also useful for analyzing the propa-gation of TE modes in rectangular waveguides filled with indefinite media [79], orthe amplification of TE evanescent modes by slabs made of such media [53].

PROBLEMS

1.1. Quasistatic plasmonic resonance of a sphere. Find the polarization of asphere of permeability 1 , 0 in the presence of a uniform quasielectrostaticfield. Assume a plasmonic type of frequency dependence 1 ¼ 10(1� v2

p=v2),

and find the frequency of oscillation of quasistatic polarization modes.

PROBLEMS 35

1.2. Quasistatic surface plasmons. Show that Laplace’s equation at the interfacebetween two media, 1 and 2, has bounded propagative solutions along suchinterface provided 11 ¼ � 12. These solutions are usually called quasistaticsurface plasmons [3].

1.3. Full-wave surface plasmons–polaritons. Find the dispersion relation for TMsurface waves at the interface between two media with 11 . 0 and 12 , 0.Show that these solutions converge to the aforementioned surface plasmons(see Problem 1.2) in the limit k ! 1. For small values of k, these waveshave a different dispersion relation and are sometimes called surface polari-tons. See [3] for a deeper discussion on surface plasmons–polaritons.

1.4. Energy flow of a plane wave in a left-handed medium. Show that equation(1.14) ensures that the density of energy (U) and the energy flux (the Poyntingvector S) of a plane wave in a lossless left-handed medium are related throughS ¼ vg U:

1.5. Left-handed concave and convex lenses. Show that a concave (convex) lensmade of a left-handed metamaterial becomes convergent (divergent). Theselenses were analyzed in [6].

1.6. Refraction of nonparaxial rays by a left-handed slab. Study the refractionof nonparaxial rays coming from a point source by a left-handed slab, andfind the distance from the source at which these rays refocus. Show that thisdistance reduces to equation (1.22) for paraxial rays.

1.7. Dispersion relation for p-polarized surface waves. Find the dispersionrelation for p-polarized (or TM, or LSM) surfaces waves at the interfacebetween a left-handed medium and free space. Show that it is deduced fromequation (1.37) by the duality transformations 1!m, m!1, E!H, H!2E(see [14]).

1.8. All-angle negative refraction by an indefinite uniaxial dielectric. Study therefraction of plane waves at the interface between free space and a uniaxialdielectric with m ¼ m0, 1k , 0 and 1? ¼ 10. Show that all-angle negativerefraction occurs for p-polarized (TM) waves when the optical axis is perpen-dicular to the interface. Similar systems have been analyzed in [9,24].

1.9. Energy flow in surface waves. Show that in the surface waves (1.37), powerflows in the backward direction inside the left-handed material, and in theforward direction in free space. Discuss, in terms of the constitutive parameters,if the whole surface wave is forward or backward.

1.10. Small resonators made with left-handed media. Study the resonant modesof a one-dimensional cavity made of two consecutive slabs of ordinary andleft-handed media, placed between two parallel metallic plates. Show thatthe condition for the resonance of TEM waves does not depend on the distancebetween the plates, but on the relative width of the slabs. This and other similarresonators and waveguides have been proposed in [47].


1.11. Modes in parallel-plate waveguides partially filled by a left-handedslab. Find, making use of the transverse transmission matrix technique, thedispersion relation for TE and TM modes in parallel plate waveguides partiallyfilled by a left-handed slab. This and other similar waveguides have been ana-lyzed in [38].

1.12. Phase compensation in single negative slabs. Show that all-angle phase com-pensation and amplification of evanescent waves also occurs in two consecutiveslabs with 0 , 12 ¼ �11 and 0 . m2 ¼ �m1, provided thewidths of both slabsare identical. Draw a plot of the amplitude of electromagnetic waves inside thissystem for the different aforementioned effects. See [52] for a deeper discussionon the properties of a pair of consecutive single negative slabs.

1.13. Asymmetric perfect lens. Find the transmission coefficient through a left-handed slab with 1 ¼ �10 and m ¼ �m0 sandwiched between free spaceand a semi-infinite medium with 0 , 1 = 10 and 0 , m = m0. This systemis sometimes called an asymmetric perfect lens [67].

1.14. Near-field near-perfect lens made with a medium with 1 ¼210. Find thetransmission coefficient for evanescent quasielectrostatic waves (i.e., solutionsto Laplace’s equation of the kind f(r) ¼ f0 exp(�jk � r)) through a slab with1 ¼ �10 and m ¼ m0. Show that this slab behaves as a near-field near-perfectlens for the quasielectrostatic field. A deeper discussion of this proposal can befound in [26] and in [68–71], among others.

1.15. TE anticutoff waveguide made with an indefinite uniaxial medium. Findthe dispersion relation for the fundamental TE1,0 mode in a rectangular wave-guide filled with a uniaxial indefinite medium with 1 ¼ 10, mx ¼ my ¼ m0,mz , 0 (see Fig. 1.9 for the axis definition). Show that this mode has ananticutoff behavior; that is, it is propagative below certain cutofffrequency and evanescent above this frequency. This device has beenstudied and experimentally demonstrated in [79].

1.16. TM anticutoff waveguide made with an indefinite uniaxial medium. Findthe dispersion relation for the fundamental TM1,1 mode in a rectangular wave-guide filled with a uniaxial indefinite medium with m ¼ m0, 1x ¼ 10,1y ¼ 1z , 0 (see Fig. 1.9 for the axis definition). Show that this mode hasan anticutoff behavior; that is, it is propagative below a certain cutoff frequencyand evanescent above this frequency. This device has been studied and exper-imentally demonstrated in [80].

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REFERENCES 41

CHAPTER TWO

Synthesis of Bulk Metamaterials

2.1 INTRODUCTION

This chapter is devoted to the design of bulk metamaterials with negative parameters.By bulk metamaterialswe mean discrete media made of a combination of unit cells ofsmall electrical size at the frequencies of interest. It will be assumed that the consti-tutive parameters of such media can be deduced from the unit cell polarizabilities byan appropriate homogenization procedure. This approach is, in fact, a direct trans-lation of the methods used in solid-state physics for the characterization of naturalsubstances. Bulk metamaterials, as they have been defined here, substantially differfrom photonic or electromagnetic crystals. In fact, the electromagnetic properties ofbulk metamaterials arise, in essence from the electromagnetic properties of its consti-tutive elements. Periodicity usually plays a secondary role in bulk metamaterials.

After the seminal works of Walter Rotman [1], John Pendry [2,3], and DavidSmith and his co-workers [4], a standard procedure was established for the designof bulk artificial media with negative parameters at microwave frequencies. This stan-dard procedure makes use of a system of metallic wires and/or plates to obtain thenegative dielectric permittivity. A system of split ring resonators (SRRs) is used toobtain the negative magnetic permeability. This standard approach will be developedthrough the main part of this chapter, in Sections 2.2 to 2.5. The homogenization ofsystems of metallic wires and plates is described in Section 2.2. In the last part of thissection, the key issue of spatial dispersion in wire media is introduced. Section 2.3 isdevoted to the synthesis of negative permeability media using SRRs. A physicalmodel for these elements, based on equivalent LC circuits, is developed. Then,Lorentz local field theory is applied to the homogenization of SRR media. Someadvanced topics, such as higher-order SRR resonances, the design of isotropicSRRs, or the behavior of SRRs at infrared and optical frequencies, will be addressedin the last part of this Section. The synthesis of left-handed metamaterials using a

Metamaterials with Negative Parameters. By Ricardo Marques, Ferran Martın, and Mario SorollaCopyright # 2008 John Wiley & Sons, Inc.

43

superposition of the aforementioned negative permittivity and negative permeabilityconfigurations will be developed in Section 2.4. Some key issues of this approach,such as the limits of the homogenization procedure in discrete left-handed meta-materials, and the justification of the superposition hypothesis underlying this pro-cedure, will be addressed at the end of the section. Finally, in the last section ofthis chapter, other proposals for the design of bulk metamaterials with negative par-ameters are described and analyzed.

It must be mentioned here that there is also an alternative and widely usedapproach for the design of metamaterials at microwave frequencies. This approachwas introduced simultaneously by Iyer and Eleftheriades [5], Caloz et al. [6], andOliner [7]. It may be classified as a nonresonant circuit analysis approach. Suchas approach substantially differs from the bulk media approach, because onlycircuit-like interactions between the nearest-neighbors elements of the system areconsidered. It also makes it possible to avoid the resonant behavior inherent inbulk design approaches. This circuit analysis approach will be described in thenext chapter. The relation between such as approach and the analysis developed inthis chapter, as well as the analysis of SRR-loaded transmission lines in the frameof the circuit analysis approach, will be developed in Chapter 3.

2.2 SCALING PLASMAS AT MICROWAVE FREQUENCIES

The simulation of plasmas at microwave frequencies became an active field ofresearch during the 1960s, mainly aimed at the simulation of radio-communicationswith spaceships during transit through the ionosphere. In a paper that became aclassic, Walter Rotmann studied the modeling of dilute plasmas—with plasma fre-quencies in the microwave range—by systems of metallic wires or plates [1].Many years later, the same problem was addressed from a different standpoint byJohn Pendry [2]. Because lossless plasmas exhibit a negative effective permittivitybelow plasma frequency, these works opened the way to the design of artificialmedia with negative dielectric permittivity. Such an approach will be developed inthis section. We will start with the simpler analysis of hollow metallic waveguidesand plates, which simulate one- and two-dimensional plasmas, respectively. Wiremedia will then be analyzed.

2.2.1 Metallic Waveguides and Plates as One- andTwo-Dimensional Plasmas

Let us consider a hollow rectangular metallic waveguide. The propagation constantcorresponding to the fundamental TE10 mode is given by

k2 ¼ v210m0 1� v2c

v2

� �, (2:1)

44 SYNTHESIS OF BULK METAMATERIALS

where vc is the cutoff frequency of the waveguide. This dispersion relation is iden-tical to that of an ideal plasma provided the cutoff frequency vc is substituted bythe plasma frequency vp. This identity suggest that a rectangular waveguide can beseen—to some extent—as a one-dimensional plasma [8] with an effective dielectricconstant of

1eff ¼ 10 1� v2c

v2

� �: (2:2)

This equivalence is supported by analysis of the wave impedance corresponding tothe considered mode [9,10],

Z ¼ vm0

k: (2:3)

In fact, if we assume that the continuous media relations

k2 ¼ v21effmeff and Z2 ¼ meff

1eff, (2:4)

are simultaneously satisfied with k and Z given by equations (2.1) and (2.3), the effec-tive dielectric constant (2.2) is deduced, as well as meff ¼ m0. Additional compu-tations show that the proper relation between the effective polarization and theaveraged electric field also holds (see Problem 2.1). Therefore, the proposed equiv-alence between rectangular waveguides and one-dimensional ideal plasmas is alsosupported by the averaging of the electromagnetic quantities. In fact, because the pre-vious discussion is not restricted to rectangular waveguides, this equivalence can beextended to any hollow waveguide supporting TE modes. In particular, in squarewaveguides the fundamental modes TE10 and TE01 are degenerated. Therefore, abunch of square waveguides such as those shown in Figure 2.1 will behave as aone-dimensional plasma for TE wave polarization.1

The above results can easily be extended to the simulation of two-dimensionalplasmas by a system of parallel metallic plates [1] such as that shown inFigure 2.1. When this structure is illuminated by a plane wave polarized with the elec-tric field parallel to the plates, and with a wavevector also parallel to them (so that theincident magnetic field is perpendicular to the plates) the TE10 mode is excited in thesystem.2 Such a TE10 mode has a propagation constant inside the system of plates asgiven by equation (2.1), and an impedance given by equation (2.3). Therefore, fromequation (2.4), the system of plates can be characterized by the effective dielectric

1It may be worth mentioning that a similar equivalence between waveguides supporting TM modes andideal magnetic plasmas can be stated (see Problem 2.2 and [11])2In fact, higher-order modes are also excited. However, it is usually supposed that only the TE10 mode has asignificant penetration inside the system, so that the effect of higher-order modes can be accounted for as aboundary condition at the interface.

2.2 SCALING PLASMAS AT MICROWAVE FREQUENCIES 45

constant (2.2), and the effective magnetic permeability meff ¼ m0. Such effective con-stants correspond to an ideal plasma with plasma frequency vp ¼ vc ¼ cp=a, wherec is the velocity of light in free space, and a is the distance between the plates.Because, for the TE10 mode, the average component of the magnetic field alongthe direction of propagation vanishes, the average fields inside the system of parallelplates (kEl and kHl) form a TEM wave, with kEl parallel and kHl perpendicular to theplates. It can also be shown (see Problem 2.1) that the average electric polarization Pinside the structure is related to the averaged electric field by

P ¼ �10v2c

v2kEl, (2:5)

which corresponds to the effective dielectric constant (2.2). In summary, the systemof parallel plates shown in Figure 2.1 behaves as a two-dimensional ideal plasma withplasma frequency vp ¼ cp=a. However, it should be kept in mind that this equival-ence is restricted to waves polarized with the electric field parallel to the plates.

FIGURE 2.1 Illustration of the development of one-, two-, and three-dimensional artificialplasma designs by using square waveguides, metallic plates, and wires. (a) A bunch ofsquare metallic waveguides behaves as a one-dimensional plasma (with a plasma frequencyequal to the cutoff frequency of the waveguides). (b) The horizontal conducting walls,which do not affect the field distribution in (a), are suppressed in order to obtain a two-dimensional design (front view). (c) The metallic boundaries of (a) are substituted by aregular mesh of wires in order to achieve an isotropic three-dimensional artificial plasma.


2.2.2 Wire Media

The designs described in Section 2.2.1 are simple and can be easily analyzed and man-ufactured. However, they are restricted to propagation in one or two dimensions, and towaves with the proper polarization. It can be guessed that a cubic mesh of conductingwires can provide the generalization of the above concepts to a three-dimensionalstructure. This concept is illustrated in Figure 2.1. If the period of the wire mesh issmaller than the free-space wavelength, it should be approximately equivalent to thebunch of waveguides shown in Figure 2.1a. Moreover, because the considered wiremesh is isotropic at large scales, it can be guessed that such system will behave asan isotropic artificial plasma for all wave polarizations and directions of propagation(at least in the long wavelength limit). The plasma frequency of such artificialplasma should be close to the cutoff frequency of the waveguide bunch vp � cp=a.

A more accurate determination of vp comes from the analysis of the equivalenttransmission line for the wire medium. Let us consider a TEM transmission lineloaded by metallic posts, as shown in Figure 2.2. This transmission line is usefulfor modelling the propagation of TEM waves along the wire media of Figure 2.1, pro-vided the electromagnetic field is polarized with the electric field parallel to a set ofwires. The per unit length series inductance of the parallel plate transmission line isLs=a ¼ m0, where the per unit length shunt capacitance is C=a ¼ 10. In the circuitmodel for the transmission line of Figure 2.2, the effect of the wires can bemodeled by periodically loading this transmission line with the shunt inductancesassociated with each metallic post. We can estimate this shunt inductance as [2]

L ’ m0a

2pln (a=r), (2:6)

FIGURE 2.2 Illustration of the transmission line model for wire media. (a) Equivalent trans-mission line (PEC: perfect electric conductor; PMC: perfect magnetic conductor). (b) Circuitmodel for the equivalent transmission line, Ls ¼ m0a, C ¼ 10a.


where r is the wire radius. In this expression, the effect of the remaining wires hasbeen taken into account by imposing that the magnetic field created by a singlewire is shielded by its nearest neighbors at a distance a from the considered wire(otherwise, this inductance will become infinite). From standard transmission linetheory it follows that the phase propagation along the wire-loaded transmissionline of Figure 2.2 is given by

k2 ¼ v2

a2(Lþ Ls)C ¼ v2m010 1� 2p

v2m010a2 ln(a=r)

� �, (2:7)

provided the periodicity a can be considered small with regard to wavelength. Bycomparing this expression with the corresponding one for an ideal plasma, thefollowing value is obtained for the plasma frequency of the wire media:

v2p ¼

2pm010a2 ln(a=r)

: (2:8)

This value coincides with the result reported in [2]. More accurate calculations [14],lead to the result

v2p ¼

2p

m010a2[ lnfa=ð2prÞg þ 0:5275]: ð2:9Þ

The reported theory on the homogenization of wire media cannot be acceptedwithout some criticism. In fact, except for extremely thin wires, the above expressionimplies that propagation of electromagnetic waves is only possible at frequencies withan associated free-space wavelength of the same order as the period of the structure.Nevertheless, near the plasma frequency, the internal wavelength is very large, so theassumption of an electrically small periodicity along the line can still be consideredvalid. However, as frequency increases, the internal wavelength soon becomes similarto the period. Needless to say, below the plasma frequency, the penetration length ofevanescent waves also soon becomes of the same order as the period. Therefore,except for a small region around the plasma frequency, the reported homogenizationseems to be against the most basic assumption of any homogenization theory: Thescale of variation of the averaged field should be much larger than the period.Nevertheless, in metamaterial analysis, the above expression is widely usedwithout too much care about its frequency range of validity. This use is ordinarilyjustified by the fact that, when other elements (split ring resonators for instance)are added to the wires, waves with an internal wavelength much larger than theperiod can propagate in the metamaterial. This argument leads, however, to the super-position problem, which will be addressed in Section 2.4.4. Ultimately, it is the agree-ment between theory and experiments that justifies the homogenization.


2.2.3 Spatial Dispersion in Wire Media

The analysis reported in the previous section is valid for TEM waves propagatingalong a set of wires, and polarized with the electric field parallel to another set ofwires (see Fig. 2.1c). From this result, and from the isotropy at large scales of thestructure, we could assume that the results of such an analysis are generalizable toany TEM wave. That is, the system of wires could be described as an ideal isotropicplasma with a plasma frequency given by equation (2.8). However, as will be shownin the following, this assumption is not always justified.

To begin, we will consider the more simple system made of a set of periodic par-allel infinite wires, as shown in Figure 2.3. For homogenized TEMwaves propagatingperpendicular to the wires and polarized with the magnetic field also perpendicular tothe wires, this medium behaves as an ideal plasma with plasma frequency given byequation (2.8). For waves with orthogonal polarization, the medium is transparentwith 1 ¼ 10. Also, for TEM waves of any polarization propagating along thewires, the medium is transparent with 1 ¼ 10 (for thin wires). However, it hasbeen shown [12] that these facts do not mean that the wire medium could be charac-terized as an uniaxial dielectric with 1x ¼ 1y ¼ 10 and 1z ¼ 10(1� v2

p=v2).

The effective permittivity tensor of the artificial medium shown in Figure 2.3 can beobtained from the analysis of the propagation of TM-to-z electromagnetic wavesthrough such medium (TE-to-z waves do not see the wires, thus being TEM waveswith propagation constant k0 ¼ v


p). The fields of such TMwaves are fully deter-

mined by the electric field component Ez, which should satisfy Hemholtz’s equation

@

@x2þ @

@y2þ g2

� �Ez ¼ 0, (2:10)

where g2 ¼ k20 � k2z (with k20 ¼ v210m0), and kz is the wavevector along the wires’axis. Boundary conditions impose that Ez ¼ 0 on the wires. Solutions to this

FIGURE 2.3 A system of perfectly conducting infinite parallel wires.


system of equations only depend on the eigenvalue g2, so the transverse dependenceof the fields is not affected by kz and v independently, but by the combination of bothg2 ¼ k20 � k2z . The eigenvalue g can be identified with the free-space wavevector, k0,corresponding to a TM wave with kz ¼ 0; g2 ¼ k20(kx, ky, 0). Therefore, the dis-persion relation for the considered TM waves must satisfy

g2 ; k20(kx, ky, kz)� k2z ¼ k20(kx, ky, 0), (2:11)

for any value of kz. The dispersion relation for such TM waves with kz ¼ 0, is known,having the plasma-like form

k2x þ k2y ¼ k20(kx, ky, 0)� k2p , (2:12)

where kp ¼ vpffiffiffiffiffiffiffiffiffiffi10m0

pis the free-space wavevector corresponding to the plasma fre-

quency vp. From this result, the dispersion relation for TM waves with kz = 0 isobtained by substitution of equation (2.11) into (2.12):

k2 ; k2x þ k2y þ k2z ¼ k20 � k2p : (2:13)

This result is not compatible with the uniaxial permittivity tensor previously assumed(see Problem 2.3). However, if the uniaxial permittivity tensor [12]

1x ¼ 1y ¼ 10, 1z(v, kz) ¼ 10 1�k2p

k20 � k2z

!; k2p ¼ v2

p10m0 (2:14)

is assumed for the parallel wire medium, then the dispersion relation (2.13) isobtained (see Problem 2.3). Thus, we can conclude that the considered wiremedium behaves as a uniaxial dielectric with the permittivity tensor (2.14). Thisresult shows that spatial dispersion, expressed in the dependence on kz of the permit-tivity tensor (2.14), plays an important role in the characterization of wire media, evenin the very large wavelength limit. Such a result may be expected, because the unitcell of the considered parallel-wire medium has an infinite extent along the z-axis,and spatial dispersion is expected to appear when the unit cell size is not smallwith regard to the wavelength [13].

The above result raises an important question on the homogenization of wiremedia: to what extent spatial dispersion should be considered in the analysis ofwire meshes? This question has been answered in [14]. The main result reportedthere is that the connected configuration shown in Figure 2.1c actually behaves asan ideal plasma, supporting homogenized TEM waves with the dispersion relation(2.1), and with an effective plasma frequency given by equation (2.9). However, aregular mesh of nonconnected wires shows strong spatial dispersion in the long wave-length limit, similar to that reported for the parallel-wire media.


2.3 SYNTHESIS OF NEGATIVE MAGNETIC PERMEABILITY

In 1852, Wilhem Weber formulated the first theory of diamagnetism, discovered byFaraday some years before. He assumed the existence of closed circuits at the molecu-lar scale, and invoked Faraday’s law to prove that currents would be induced in thesecircuits when they were under the effect of an external time-varying magnetic field.As the secondary magnetic flux created by such currents would be opposite to thatcreated by the external field, this mechanism would explain the diamagnetismreported by Faraday [15]. In fact, it is easily shown that the magnetic polarizabilityof a lossless conducting ring along its axis (namely the z-axis) is given by

mz ¼ ammzz Bext

z and ammzz ¼ �p2r4

L, (2:15)

where ammzz is the polarizability,3 Bext

z is the z-component of the applied magneticfield, r is the radius of the ring, and L the self-inductance of the ring. However,the diamagnetic effect associated with a closed metallic ring is not strong enoughto produce negative values for m. In fact, the self-inductance of a perfect conductingring can be efficiently calculated as [13]

L ¼ m0r ln16rd

� �� 2

� �, (2:16)

where d is the diameter of the wire. This expression gives L � m0r for any realisticr/d ratio. Thus, m0jamm

zz j � p2r3. Therefore, the magnetic susceptibility of an

arrangement of rings (which can be estimated as xm � m0ammzz =V , where V & (2r)3

is the volume per ring) is jxj . p2=8 � 1. That is, it does not seem possible toobtain an effective negative m from any arrangement of closed metallic rings.

The magnetic polarizability of a closed metallic loop can be easily enhanced byloading the loop with a capacitor. This configuration, proposed by Schelkunoff[16], provides the following expression for amm

zz :

ammzz ¼ p2r4

L

v20

v2� 1

� ��1

, (2:17)

where v0 ¼ 1=ffiffiffiffiffiffiLC

pis the resonant frequency of the LC circuit formed by the loop

and the capacitor. This expression shows that, just above the frequency of resonance,the polarizability becomes negative and very large. Therefore, it is expected that aregular array of capacitively loaded metallic loops will show a negative magnetic

3The repeated sub-index zz indicates that it is due to a force directed along the z-axis, and produces an effectalong the same axis. The repeated super-index mm indicates that the external force is magnetic, and that theeffect is also magnetic.

2.3 SYNTHESIS OF NEGATIVE MAGNETIC PERMEABILITY 51

permeability just above the frequency of resonance of the loops. However, thisconfiguration may be difficult to manufacture at microwave and higher frequencies,a fact that is crucial if we are interested in the design of effective media that mayinvolve hundreds or, perhaps, thousands of elements. This difficulty is solved atmicrowave frequencies if the lumped capacitance is substituted by a distributedone. This modification of Shelkunoff’s proposal leads to the split ring resonator(SRR) proposed by Pendry in [3]. Using this design it is possible to manufacturelarge series of small loops, of very high and negative magnetic polarizability, byusing standard planar photo-etching techniques. The analysis of Pendry’s SRR andrelated geometries is the main purpose of this section. This analysis will accountfor the main properties of these elements, providing accurate analytical design for-mulas useful for the design of SRR-based metamaterials.

In the last part of this section some related topics will be addressed. Such topicsinclude the design of isotropic SRRs useful for isotropic metamaterial design, aswell as the behavior of split ring resonators when they are scaled down in order toobtain a resonant magnetic response at infrared and optical frequencies.

2.3.1 Analysis of the Edge-Coupled SRR

The SRR (in the following, “edge-coupled SRR”, or EC-SRR), as it was initially pro-posed by Pendry [3], consists of two concentric metallic split rings, printed on amicrowave dielectric circuit board (Fig. 2.4). When it is excited by a time-varyingexternal magnetic field directed along the z-axis, the cuts on each ring (which areplaced on opposite sides of the EC-SRR) force the electric current to flow fromone ring to another across the slots between them, taking the form of a strong displa-cement current. The slots between the rings therefore behave as a distributed capaci-tance, and the whole EC-SRR has the equivalent circuit shown in Figure 2.5b, whereL is the EC-SRR self-inductance and C is the capacitance associated with each EC-SRR half. This capacitance is C ¼ prCpul, where r is the mean radius of the EC-SRR(r ¼ rext � c� d=2), and Cpul is the per unit length capacitance along the slotbetween the rings. The total capacitance of this circuit is the series connection of

FIGURE 2.4 The edge-coupled split ring resonator (EC-SRR). Metallizations are in whiteand dielectric substrate in gray.


the capacitances of both EC-SRR halves, that is C/2. Thus, neglecting losses, theequation for the total current I on the circuit is given by

2jvC

þ jvL

� �I ¼ S, (2:18)

where S is the external excitation. The reported circuit model was first proposed in[17]. It is valid as long as the perimeter of the ring can be considered small withregard to a half-wavelength, and the capacitance associated with the cuts on eachring can be neglected. Under such assumptions, the currents on each ring mustvanish at the cuts, and the angular dependence of the currents on each ring can beassumed to be linear (so that the total current on both rings is constant). Such assump-tions also imply that the voltage across the slots is constant in both EC-SRR halves(see Fig. 2.5c and d for a detailed illustration of these hypotheses). A more detailedcircuit model, which takes into account the gap capacitance and includes atransmission-line model for the slot between the rings has been reported in [18]. Ithas been shown that such a model converges to the previous model when the capaci-tance of the cuts on each ring is neglected, and the electrical length of the EC-SRR issmall [18]. As both approximations are usually fulfilled for any practical EC-SRRdesign, the more simple model developed above will be assumed valid throughoutthis book. The frequency of resonance, v0, of the EC-SRR can be obtained fromthe proposed model by solving equation (2.18) for S ¼ 0. This leads to

v20 ¼

2LC

¼ 2prCpulL

: (2:19)

As has already been mentioned, near-resonance current lines flow across the slotbetween the rings as displacement current lines. Thus, the EC-SRR self-inductance,

FIGURE 2.5 Quasistatic circuit model for the EC-SRR. (a) Sketch of the resonator. (b)Equivalent circuit for the determination of the frequency of resonance, where C is the capaci-tance across the slots on the upper and lower halves of the EC-SRR: C ¼ prCpul. (c) Plots ofthe angular dependence of currents on the inner ring (dashed line), on the outer ring (dash–dotted line), and of the total current on both rings (solid line). (d ) Plots of the angular depen-dence of the voltage on the inner ring (dashed line) and on the outer ring (dash–dotted line).


L, can be modeled as the inductance of an average ring of radius r (the mean radius ofthe ring) and width c (the width of each ring) [17]. The analysis in [18] leads to thesame conclusion: The self-inductance L in equation (2.19) must be the average of theinductances of both rings (without the gaps). In [19], approximate expressions for theaverage inductance L, based on a variational calculation, are provided. Closedexpressions for the per unit length capacitances between the rings can be found inmany microwave handbooks (see, for instance, [20]). Such explicit expressions forL and Cpul are given in the Appendix.

If the EC-SRR is excited by an external magnetic field, the excitation in equation(2.18) is S ¼ �jvFext, where Fext is the external magnetic flux across the EC-SRR.The equation for the total current becomes

I ¼ Fext

L

v20

v2� 1

� ��1

: (2:20)

As has been shown, the EC-SRR essentially behaves as a capacitively loaded con-ducting loop. It therefore exhibits a resonant magnetic polarizability amm

zz , given byequation (2.17).

A careful consideration of the behavior of the EC-SRR shows that, near the reson-ance, the particle reacts to an external magnetic field not only as a strong magneticdipole, but also as a strong electric dipole. In fact, when the EC-SRR is excited atresonance, charges in the upper half of the EC-SRR must be the images of chargesat its lower half, as sketched in Figure 2.5a. Two parallel electric dipoles directedalong the y-axis are generated at each EC-SRR half. From the well-known Onsagersymmetry principle [21], it directly follows that the EC-SRR resonance can also beexcited by an external electric field directed along the y-axis of Figure 2.5a. Inboth cases, a strong magnetic dipole along the z-axis, and a strong electric dipolealong the y-axis, are simultaneously excited. Moreover, the EC-SRR exhibits somenonresonant electric polarizabilities along the x- and y-axis, which can be approxi-mated by those of a metallic disk of radius rext ¼ r þ d=2þ c [10]. All theseresults can be summarized by the following set of equations [19]:

mz ¼ ammzz Bext

z � aemyz E

exty (2:21)

py ¼ aeeyyE

exty þ aem

yz Bextz (2:22)

px ¼ aeexxE

extx (2:23)

where the symmetry of the cross-polarizabilities derived from the Onsager symmetryprinciple [21,22], (aem

yz ¼ �amezy ) has been explicitly introduced.

The EC-SRR magnetic polarizability is directly obtained from equation (2.20)by taking into account that the dipolar magnetic moment of the ring is mz ¼ pr2I,


and Fext ¼ pr2Bextz ; that is,

mz ¼p2r4

L

v20

v2� 1

� ��1

Bextz , (2:24)

a result that coincides with equation (2.17). The electric dipole induced by the exter-nal field component Bext

z can be computed as

py ¼ 2ðp0prr sinf df ¼ 4rpr, (2:25)

where pr is the radial per unit length electric dipole created along the slot between therings in the upper half of the EC-SRR ( y . 0 in Fig. 2.5a). This quantity can bewritten as pr ¼ qdeff , where q is the per unit length charge on the outer ring (thecharge on the inner ring must be equal in magnitude and of opposite sign), anddeff some effective distance deff ’ cþ d. That is,

pr ¼ qdeff ¼ CpulVdeff , (2:26)

where V is the voltage difference across the slot between the outer and the inner rings,which is a constant in the considered approximation (Fig. 2.5d). This voltage can becalculated from 2V ¼

ÞE � dl, where the field integral is taken along a path going on

the rings, and passing from one ring to another across the slots. From Faraday’s law,2V ¼ �jv(LI þFext). Therefore, taking into account equations (2.25) and (2.26),

py ¼ �2jvrCpuldeff (LI þFext), (2:27)

and taking into account equation (2.20) and Fext ¼pr2Bextz ,

py ¼ �2jpr3deffCpulv20

v

v20

v2� 1

� ��1

Bextz : (2:28)

Let us now consider the behavior of the EC-SRR under an electric excitation. Theexternal excitation S is now the series connection of the external voltage across thecapacitors formed by the upper and the lower EC-SRR halves. This externalvoltage can be estimated as two times the average voltage created by the externalfield through each EC-SRR half.

S ¼ 2kV extl ¼ 2p

ðp0Eexty deff sinf df ¼ 4

pdeffE

exty : (2:29)


By substitution in equation (2.18), the total current in the EC-SRR is found to be

I ¼ j4deffpvL

v20

v2� 1

� ��1

Eexty : (2:30)

This current creates a magnetic moment mz ¼ pr2I, which, taking into accountequation (2.19), can be written as4

mz ¼ 2jpr3deffCpulv20

v

v20

v2� 1

� ��1

Eexty : (2:31)

Moreover, when the EC-SRR is under this excitation, the resonant current generatedaround the rings must also create an electric moment, due to the radial polarization(2.26) along the slots. It is easier to obtain the per unit length charge on the outerring from the current on this ring, and from charge conservation,

jvq ¼ � 1r

dIoutdf

, (2:32)

where Iout is the current on the outer ring. As, according to the model sketched inFigure 2.5, the dependence of Iout on f is linear, and Iout takes a maximum Iout ¼I at f ¼ 0, and a minimum Iout ¼ 0 at f ¼ p, the derivative in equation (2.32) canbe evaluated as dIout=df ¼ �I=p. Therefore, taking into account equation (2.30),

q ¼ I

jvpr¼ 4deff

p2v2rL

v20

v2� 1

� ��1

Eexty : (2:33)

The total electric dipole associated with the EC-SRR is evaluated by substitution inequations (2.25) and (2.26). Taking into account equation (2.19), this electric dipolecan be written as

py ¼ 4d2eff r2C2

pulLv20

v

� �2v20

v2� 1

� ��1

Eexty : (2:34)

Finally, the EC-SRR must show a nonresonant electric polarizability in theplane of the rings, which can be approximated as the electric polarizability of ametallic disk of the same size as the EC-SRR. This polarizability is

4Note that the symmetry of the cross-polarizabilities assumed in equations (2.21) and (2.22) is fulfilled.


a ¼ 1016r3ext=3, where rext ¼ r þ 2cþ d [10]. In summary, the polarizabilities of theEC-SRR are as follows:5

ammzz ¼ p2r4

L

v20

v2� 1

� ��1

(2:35)

aeexx ¼ 10

163r3ext (2:36)

aemyz ¼ �ame

zy ¼ �2jpr3deffCpulv20

v

v20

v2� 1

� ��1

(2:37)

aeeyy ¼ 10

163r3ext þ 4d2eff r

2C2pulL

v20

v

� �2v20

v2� 1

� ��1

: (2:38)

In such equations the effect of the polarization charges induced in the dielectric sub-strate has been neglected. However, they should affect the electric polarizabilities. Ifthe substrate is thin (t � d), it can be assumed that the polarization charges inducedon each side of the dielectric substrate cancel each other. Therefore, the electric dipole(2.26) is mainly due to the free charge alone, and the above expressions are approxi-mately valid. However, for thick substrates, it is the overall (free and polarization)charge that contributes in equation (2.26). In such a case it can be more appropriateto use the in vacuo per unit length capacitance between the rings, C0

pul, instead of Cpul

in equation (2.26).6 Moreover, it is also more appropriate to use

S ¼ 2kV extl ¼C0pul

Cpul

2p

ðp0Eexty deff sinf df ¼

C0pul

Cpul

4pdeffE

exty , (2:39)

instead of equation (2.29).7 All these changes are accounted for by simply substitut-ing Cpul with C0

pul in equations (2.35) to (2.38) [17,19] (of course, equation (2.19)

should not be changed).From equations (2.35) to (2.38) it follows an interesting property of the resonant

part of the polarizabilities ammzz , aem

yz , and ammyy , that is,

ammzz aee

yy � 10163r3ext

� �¼ �(aem

yz )2: (2:40)

5The value for ayyee slightly differs from the value reported in [17]. We are in debt to Dr Juan Baena and Dr

Lukas Jelinek for bringing this mistake to our attention.6In other words, the total (free plus polarization) charge q0 ¼ C0

pul=Cpulq is used instead of the freecharge q.7The reason for this choice is that the voltage induced by an external electric field Eext in a capacitor par-tially filled by a dielectric is approximately given by C0=C Eextd, where d is the distance between the plates.An additional reason is that this choice guarantees that the cross-polarizabilities satisfy the Onsager sym-metry: ayz

em¼2azyme.


In fact, this property is quite general and comes from the quasistatic nature of the firstEC-SRR resonance, and in particular from the possibility of modeling this resonance,by an LC circuit.

Up to now, we have considered a lossless EC-SRR. However, as was already men-tioned in Chapter 1, losses play an important role in the characterization of meta-materials. As SRRs are resonant elements, it can be expected that metamateriallosses will be mainly associated with ohmic losses in the SRRs. These losses caneasily be taken into account by considering the AC resistance of the SRRs, whichcan be estimated as [19]

R ¼

2pr0chs

if h=2 , d,

pr0cds

otherwise,

8><>: (2:41)

where h is themetal thickness,s is the conductivityof themetallic strips, and d is the skindepth. The introduction of this resistance in the above analysis leads to the substitution

L ! L ¼ Lþ R

jv: (2:42)

Such a substitution must be made through equations (2.35) to (2.38) (as well as inequations (2.52) and (2.53) in the next section). It implies that the frequency ofresonance (2.19) must also be substituted by the function

v20(v) ¼

2

LC¼ 2

prCpulL: (2:43)

A straightforward calculation leads to

v20

v2� 1

� ��1

¼ v20

� �1 v2v20

v20 � v2 þ jvg

, (2:44)

where g ¼ R=L. After the substitutions L ! L and v0 ! v0 in equations, (2.35) to(2.38), and taking into account equation (2.44) as well as the relation Lv2

0 ¼ Lv20, the

polarizabilities for a lossy EC-SRR take the form

ammzz ¼ p2r4

L

v2

v20 � v2 þ jvg

� �, (2:45)

aeexx ¼ 10

163r3ext, (2:46)

aemyz ¼ �ame

zy ¼ �2jpr3deffCpulv20

v

v2

v20 � v2 þ jvg

� �, (2:47)


and

aeeyy ¼ 10

163r3ext þ 4d2eff r

2C2pulL

v20

v

� �2v2

v20 � v2 þ jvg

� �: (2:48)

These equations have a quite familiar form, showing the typical Lorentzian form nearthe resonance. However, they are not causal for high values of v, due to the presenceof powers of v in the numerator. Nevertheless, this fact has no importance, becausethe behavior of the polarizabilities at high frequencies is actually determined byhigher-order resonances, which are not accounted for in equations (2.45) to (2.48).

The frequency of resonance of an EC-SRR can be measured by placing the EC-SRR inside a rectangular waveguide and measuring the transmission coefficient,which must show a sharp dip at the EC-SRR resonance [19]. These experiments sys-tematically show an agreement with the circuit model developed in this section withina few percent of error. The presence of cross-polarization effects in the EC-SRR wasexperimentally checked in [23] (and previously in [24]) by measuring the trans-mission coefficient through a waveguide loaded with an EC-SRR placed at differentorientations inside the waveguide (Fig. 2.6). As can be seen, the aforementioned dipin the transmission coefficient appears or not depending on the particle orientation.Only in orientation 4 of Figure 2.6 is there neither magnetic flux across theEC-SRR, nor an electric field component along the y-axis of the particle.According to equations (2.35) to (2.38), it is only for this orientation that the reson-ance should not appear, as can be seen in the figure.

The reported expressions for the EC-SRR polarizabilities in equations (2.35) to(2.38) can be used for the computation of the effective permeability and permittivityof artificial magnetic media made from periodic or nonperiodic arrangements ofEC-SRRs. According to the above set of equations, an electric permittivity tensorand a magnetic susceptibility tensor do not suffice for the complete characterizationof these media. They are, in general, bianisotropic, and a cross-polarization tensormust also be included for their complete characterization.

2.3.2 Other SRR Designs

In the previous section, the EC-SRR originally proposed by Pendry [3] was analyzed.It was shown that this element may produce a strong magnetic polarizability near itsresonance. Because the EC-SRR is electrically small at resonance, and it can be easilyand reliably manufactured by using standard photo-etching techniques, its usefulnessfor the design of magnetic metamaterials is apparent. However, some properties ofthe EC-SRR, such as bianisotropy, may be the origin of unwanted effects in the meta-material. Moreover, the EC-SRR electrical size, although small, cannot be reduced inpractice to values much smaller than l/10, where l is the free-space wavelength. Thislimitation comes from the edge-coupling of the metallic rings forming the EC-SRR.This fact implies a logarithmic dependence with the inverse of the distance betweenthe rings, d, for the per unit length capacitance Cpul. Thus, Cpul cannot be increasedtoo much by reducing d (at least for practical values of d ). Therefore, the frequency of


resonance (2.19) cannot be made too small. Such limitations can be overcome bymeans of small modifications of the EC-SRR design. This section is devoted to abrief description of such alternatives.

2.3.2.1 The Broadside-Coupled SRR The broadside-coupled SRR (BC-SRR)was proposed in [17] in order to avoid the EC-SRR bianisotropy. It has the additionaladvantage of a potentially much smaller electrical size [19]. The BC-SRR is shown inFigure 2.7. The main modification with regard to the EC-SRR is that both rings areprinted at both sides of the dielectric board, so that the per unit length capacitanceCpul is the capacitance of the broadside-coupled strips. This modification does notsubstantially affect the behavior of the resonator. Therefore, the equivalent circuit

FIGURE 2.6 (a) Experimental determination of the frequency of resonance, and demon-stration of cross-polarization effects in the EC-SRR. Position 1: electric and magnetic exci-tation. Position 2: magnetic excitation only. Position 3: Electric excitation only. Position 4:no excitation. (b) Behavior of the nonbianisotropic SRR (NB-SRR) under the same excitations.The absence of electric excitation (3 and 4) shows the absence of cross-polarization effects.(Source: Reprinted with permission from [23]; copyright 2005 by the IEEE.)


for the BC-SRR, as well as the current and voltage distributions, are the same as forthe EC-SRR (Fig. 2.5b–d ). As in the EC-SRR, near the resonance, charges in theupper half of the BC-SRR are the images of charges in the lower half. However,as is sketched in Figure 2.7, this charge distribution does not result in a net electricdipole. Thus, the BC-SRR is nonbianisotropic. The same conclusion is obtained fromsymmetry considerations: The BC-SRR has inversion symmetry with regard to thecenter of both rings. Therefore, any second-order pseudo-tensor characterizing theresonator (for instance, the cross-polarizability tensor) must vanish.

From the above considerations, it turns out that the BC-SRR polarizabilities aresummarized by the following set of equations [19]:

mz ¼ ammzz Bext

z (2:49)

py ¼ aeeyyE

exty (2:50)

px ¼ aeexxE

extx (2:51)

where

ammzz ¼ p2r4

L

v20

v2� 1

� ��1

(2:52)

and

aeexx ¼ aee

yy ¼ 10163r2ext, (2:53)

where v0 is the frequency of resonance, which can still be obtained from equation(2.19). However, Cpul in equation (2.19) is now the per unit length capacitancebetween the broadside-coupled rings forming the BC-SRR (closed expressions forthis parameter are given in the Appendix).

The per unit length capacitance for the BC-SRR approximately corresponds to aparallel plate capacitor. Therefore, it has a linear dependence with the inverse of

FIGURE 2.7 Broadside-coupled split ring resonator (BC-SRR).


the width t of the dielectric substrate. Thus, the frequency of resonance varies as thesquare root of t. It also varies as the inverse of the square root of the dielectric per-mittivity of the substrate. Both properties imply that the electrical size at resonanceof the BC-SRR can be made potentially much smaller than for the EC-SRR. Inorder to achieve such a goal, thin substrates of high permittivity can be used. Thisbehavior is illustrated in Figure 2.8, where the electrical sizes at resonance of EC-and BC-SRRs of similar characteristics are compared.

2.3.2.2 The Nonbianisotropic SRR The nonbianisotropic SRR (NB-SRR) wasinitially proposed in [25] in order to avoid EC-SRR bianisotropy while keeping auniplanar design. This structure is shown in Figure 2.9. It can be easily observedthat the equivalent circuit, as well as the frequency of resonance of this element,should be the same as for an EC-SRR with similar dimensions. However, the NB-SRR, like the BC-SRR, has inversion symmetry with regard to its center.Therefore, the NB-SRR polarizabilities are formally given by the same set ofequations as the BC-SRR (2.49–2.53). The frequency of resonance is given byequation (2.19), with Cpul being the same as for the EC-SRR. An experimentaldemonstration of the absence of cross-polarization effects in the NB-SRR is shownin Figure 2.6.

2.3.2.3 The Double-Split SRR An alternative way to obtain inversion sym-metry, thus overcoming bianisotropy, is by introducing additional cuts in the EC-SRR design. The double-split SRR (2-SRR) [25] shown in Figure 2.10 is anexample of this strategy. The equivalent circuit for the 2-SRR is also shown inFigure 2.10. As the capacitances in this equivalent circuit are C ¼ prCpul=2, thetotal capacitance of the circuit is four times smaller than for the conventional EC-SRR. Therefore, the frequency of resonance of the 2-SRR is twice the frequencyof resonance of an EC-SRR or NB-SRR of the same size and shape:

v0[2-SRR] ¼ 2v0[EC-SRR] ¼ 2v0[NB-SRR]: (2:54)

Because it implies a larger electrical size at resonance, this property is a cleardisadvantage of the 2-SRR. However, the high symmetry of the 2-SRR will makethis design very useful for the development of isotropic negative magnetic per-meability media (see Section 2.3.5). As the 2-SRR is a nonbianisotropic design,the 2-SRR polarizabilities are still formally given by equations (2.49) to (2.53).

2.3.2.4 Spirals Spirals are well-known resonators in planar microwave circuitry.Their usefulness for the design of negative magnetic permeability and left-handedmedia was shown in [26]. As an example, we will analyze the two-turns spiral reso-nator (2-SR) shown in Figure 2.11. A quasistatic analysis of this configuration leadsto the equivalent circuit, and to the current and voltage distributions shown in thefigure. This design also provides a strong magnetic dipole at resonance, thus beinguseful for metamaterial design. From the equivalent circuit of Figure 2.11, it


FIGURE 2.8 Frequency of resonance and normalized electrical size (2rext/l) for severalEC-SRRs (a) and BC-SRRs (b) with the same external radius rext ¼ 0.6 mm and ring widthc ¼ 0.2mm, printed on several dielectric substrates. (Source: Reprinted with permissionfrom [19]; copyright 2003, IEEE.)


follows that the frequency of resonance of the 2-SR must be half the frequency ofresonance of the EC-SRR—or the NB-SRR—of the same size and shape:

v0[2-SR] ¼12v0[EC-SRR] ¼

12v0[NB-SRR]: (2:55)

This property is a clear advantage, as it implies a smaller electrical size at resonance.The electrical size can still be reduced by increasing the number of turns [26] (seeProblem 2.4).

Relations (2.54) and (2.55) have been experimentally checked. Table 2.1 shows anexample of such results [25].

From the current and charge distributions on the 2-SR at resonance (Fig. 2.11) itcan be guessed that this design will be also nonbianisotropic. However, this con-clusion comes only from the quasistatic analysis and not from the symmetries ofthe element. In practice, any 2-SR design must imply long strips (of length �4pr,where r is the mean radius of the 2-SR). These strips may be long with regard tothe half-wavelength at resonance. Thus, the reported quasistatic analysis, althoughaccurate enough for the determination of the frequencies of resonance, may notgive accurate results for the polarizabilities. As a consequence, the 2-SR always

FIGURE 2.9 Nonbianisotropic split ring resonator (NB-SRR). Metallizations are in whiteand dielectric substrate in gray.

FIGURE 2.10 Sketch of a double-split SRR (a) and its equivalent circuit (b). The four capa-citances C of the equivalent circuit correspond to the capacitances across the four slots betweenthe rings C ¼ prCpul/2.


presents some degree of bianisotropy, although smaller than for the EC-SRR [27].This result shows the importance of symmetry in order to avoid bianisotropy: Onlydesigns invariant by inversion, or by other appropriate symmetry, are completelyfree of this effect.

2.3.3 Constitutive Relationships for Bulk SRR Metamaterials

Theeffectiveconstitutiveparameters ofbulkSRRmediacanbe found fromthepolariza-bilities obtained in previous sections. There are many possible periodic or nonperiodiccombinations of SRRs that provide an effective medium. In fact, the only necessarycondition is that the size of the unit cell must be much smaller than the internal wave-length. In this section, a cubic periodic arrangement of coplanar SRRs will be con-sidered in order to illustrate the general procedure. The resulting artificial medium,which is illustrated in Figure 2.12, is strongly anisotropic. Moreover, due to the pre-sence of cross-polarizabilities in the SRRs (see Section 2.3.1), the resulting mediumis also bianisotropic. Therefore, the most general linear constitutive relationships must

FIGURE 2.11 (a) Sketch of a two-turns spiral resonator (2-SR). (b) Equivalent circuit for thedetermination of the frequency of resonance, where C is the capacitance across the slot betweenthe rings: C ¼ 2prCpul. (c) Angular dependence of current along the inner (dashed line) andouter (dot–dashed lined) strips. (d ) Voltage along the inner (dashed line) and outer (dot–dashed line) strips. Such distributions (c and d ) imply a constant voltage across the slot anda total current (the sum of currents on both strips) constant with f (solid line in (c)).

TABLE 2.1 Comparison Between the ResonanceFrequency Provided by Experiment (exp) and Theory(th) for an SRR, a 2-SRR, and a 2-SR of SimilarDimensions Printed on the Same Substrates

f0exp (GHz) f0

th (GHz)

SRR 5.03 5.062-SSR 10.31 10.132-SR 2.66 2.53

The external radius of the three particles is 2.54mm, the width of thestrips is of 0.19mm and its separation 0.17mm. The substrate is0.49mm thick with a permitivitty of 2.4310 [25].


be considered [22]:

D ; 10Eþ P ¼ 10(1þ x¼e) � Eþ j


pk¼ �H (2:56)

B ; m0(HþM) ¼ �jffiffiffiffiffiffiffiffiffiffi10m0

p(k¼)T � Eþ m0(1þ x

¼m) �H, (2:57)

where P andM are the polarization and magnetization vectors, (.)T indicates the trans-pose, and x

¼e, x

¼m, and k

¼are the electric, magnetic, and magnetoelectric susceptibility

tensors, which are related to the permittivity and permeability tensors through

1¼ ¼ 10(1þ x

¼e ); m

¼ ¼ m0(1þ x¼m): (2:58)

In equations (2.56) and (2.57), the symmetry properties of the generalized suscepti-bilities [21], derived from the Onsager principle, have been explicitly included. Thesesymmetries also imply that x

¼e and x

¼m must be symmetric tensors, so that the whole

medium is bianisotropic and reciprocal [22].The simplest approach for the computation of the constitutive tensors x

¼e, x

¼m, and

k¼is to ignore couplings between adjacent elements. In such an approach, each sus-

ceptibility is simply the corresponding polarizability, divided by the volume of theunit cell. In this zero-order approach, we obtain

x¼e ¼

110a3

a¼ee, x

¼m ¼ m0

1a3

a¼mm, and k

¼ ¼ �j

ffiffiffiffiffiffim0

10

r1a3

a¼em, (2:59)

FIGURE 2.12 Artificial magnetic medium made of a cubic three-dimensional array of SRRs(EC-SRRs in the figure).


where a¼ee, a

¼mm, and a¼em are the polarizability tensors of the SRRs, and a is the lattice

parameter. For the artificial medium of Figure 2.12, only the susceptibilities

xe,xx ¼1

10a3aeexx; xe,yy ¼

110a3

aeeyy; xm,zz ¼

m0

a3ammzz ;

kyz ¼ �j


10

r1a3

aemyz (2:60)

are different from zero.A better approximation, which takes into account the coupling between elements

in a rather simple way, makes use of the well-known Lorentz local field theory (see[9], for instance). For a cubic array, this approximation considers that each element isexcited by a local electric field Eloc ¼ Eþ P=310, and by a local magnetic fieldHloc ¼ HþM=3 (or, equivalently, Bloc ; m0Hloc ¼ B� (2=3)m0M). When thisapproach is applied to the coupled electric and magnetic polarizations, Py and Mz,it leads to the equations

Mz ¼1a3

m0ammzz Hz þ

Mz

3

� �� aem

yz Ey þPy

310

� �� , (2:61)

Py ¼1a3

aeeyy Ey þ

Py

310

� �þ m0a

emyz Hz þ

Mz

3

� �� : (2:62)

At this point it may be convenient to rewrite expressions (2.45) to (2.48) in the form

ammzz ¼ amX

�1; am ¼ p2r4

L(2:63)

aeexx ¼ a0; a0 ¼ 10

163r3ext (2:64)

aemyz ¼ �ame

zy ¼ aemX�1; aem ¼ �2jpr3deffCpul

v20

v(2:65)

aeeyy ¼ a0 þ aeX

�1; ae ¼ 4d2eff r2C2

pulLv20

v

� �2

, (2:66)

where

X ¼ v20=v

2 � 1þ jg=v�

: (2:67)

With such definitions, equation (2.40) takes the form

amae ¼ �a2em: (2:68)


The next step is to solve equations (2.61) and (2.62) for Py and Mz. After a cumber-some but straightforward calculation, and taking equation (2.68) into account, it isfound that

Py ¼1D

LEy þm0aem

a3Hz

n o(2:69)

and

Mz ¼1D

�aem

a3Ey þ K

m0am

a3Hz

n o, (2:70)

where

D ¼ Kv20

v2� 1þ ae

3K10a3þ m0am

3a3

� �þ j

g

v

� �, (2:71)

L ¼ a0

a3v20

v2� 1� ae

a0þ m0am

3a3

� �þ j

g

v

� �, (2:72)

and

K ¼ 1� a0

310a3¼ 1� 16r3ext

9a3: (2:73)

Finally, from the above equations it directly follows that

xe,yy ¼L

10D; xm,zz ¼

K

D

m0am

a3; kyz ¼ �j


10

raem

Da3: (2:74)

The main difference between these results and equation (2.60) is a variation of thefrequency of resonance, from v0 to v1, given by

v21 ¼ v2

0 1þ ae

3K10a3þ m0am

3a3

� ��1

: (2:75)

Bianisotropy has relevant effects in wave propagation through the considered SRRmedia. The general dispersion relation for plane waves of arbitrary polarization israther complicated8, so we will only consider some specific examples. Plane wavespropagating along the x-axis of Figure 2.12, with the electric field polarized alongthe y-axis (plane waves with the orthogonal polarization see a medium with 1 ¼ 10

8An implicit expression for this dispersion relation can be found in [28].


and m ¼ m0) are TEM waves with the dispersion relation [17]

kx ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimzz1yy � m010k

2yz

q: (2:76)

However, plane waves propagating in the y direction with the electric field parallel tothe x-axis are TM waves with the E field elliptically polarized in the x–z plane. Thedispersion relation for such waves is

k ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffi1yymeff

pand meff ¼ mxx � 10m0

k2yx

1yy(2:77)

(see Problems 2.7 and 2.9).The constitutive parameters for the considered medium when the EC-SRRs are

substituted with nonbianisotropic inclusions (such as BC-SRRs or NB-SRRs) canbe obtained from the above expressions by taking aem, ae ¼ 0. It is apparent thatbianisotropy disappears in such media, which can be included into the category ofindefinite media [29], already analyzed in Chapter 1.

The behavior of the relevant constitutive parameters in a typical cubic array(Fig. 2.12) of EC-SRRs and BC-SRRs is illustrated in Figure 2.13 andFigure 2.14, respectively. Lorentz local field theory is used for the computation ofthe constitutive parameters in both cases. Because the EC-SRR medium isbianisotropic, there are four relevant parameters in Figure 2.13: magnetic per-meability mzz, magneto-electric susceptibility kyz, and the dielectric permittivities

FIGURE 2.13 Constitutive parameters of the cubic array of Figure 2.12. The Lorenzapproximation (2.74) was used for the modeling of the coupling between elements. TheMedium parameters are rext ¼ 1.24 mm, c ¼ 0.2 mm, d ¼ 0.1mm, and 1 ¼ 10. The metal iscopper with a thickness of 35mm. Periodicity is a ¼ 2.5mm.


1yy and 1xx. Except for this last quantity, all the remaining constitutive parametersshow a resonant behavior, and take negative values above the resonance. Unlikethe EC-SRR medium, the BC-SRR medium is not bianisotropic (see Section2.3.2). Therefore, only two parameters are relevant for this structure: the magneticpermeability mzz, and the dielectric permittivities 1yy ¼ 1xx. Moreover, only one ofthese quantities—the magnetic permeability mzz—is resonant, taking negativevalues above the resonance. In order to evaluate the importance of the differenthomogenization procedures for SRR media, the magnetic permeability for theBC-SRR medium obtained from the zero-order approach (2.60) is also shown inFigure 2.14. Except for a small shift in the frequency of resonance, the results arequite similar. Therefore, we can conclude that the zero-order approach is useful, atleast as a rough approximation, for SRR media characterization.

2.3.4 Higher-Order Resonances in SRRs

So far, only the first resonance of the SRRs has been analyzed. For such resonance itwas shown that the total current excited on the SRR is almost uniform; so that thewhole SRR behaves as a small closed loop of uniform current, with a strong associ-ated magnetic moment. As was shown, such resonance can be modeled by an LCcircuit model, whose L and C values can be extracted from the SRR geometry. Forthis reason, the first SRR resonance has been named the quasistatic resonance[30]. SRRs also exhibit higher-order resonances that can be excited at higher frequen-cies. The analysis of such resonances is of interest because they can affect the beha-vior of the metamaterial at high frequencies. They can be also useful for the design of

FIGURE 2.14 Real (solid lines) and imaginary (dashed lines) parts of the constitutive par-ameters of the cubic array of Figure 2.12 when BC-SRRs are used as basic elements. Themedium parameters are rext ¼ 1.24mm, c ¼ 0.2mm, t ¼ 0.3mm, and 1 ¼ 10. The metal iscopper with a thickness of 35mm. Periodicity is a ¼ 2.5mm. (a) Lorenz approximation(2.74). (b) Zero-order approximation (2.59).


some microwave devices [31]. Distributed circuit models to describe SRRhigher-order resonances, based on the assumption that almost all the electric fieldlines go across the slots between the rings, have been recently proposed [18]. Fromsuch circuit models, as well as from experiments and simulations [30], it followsthat strongly nonuniform total currents are excited on the SRR at higher-orderresonances. Therefore, such resonances will be referred to as dynamic resonances.

It is illustrative to compare the behavior of higher-order resonances in anSRR presenting cross-polarization effects (for instance the EC-SRR of Fig. 2.5),and in an SRR with inversion symmetry, where such effects are not present (forinstance the BC-SRR of Fig. 2.7 or the NB-SRR of Fig. 2.9). This analysis hasbeen developed in detail in [30] on the basis of numerical simulations and experi-ments. In this section we will briefly describe the main conclusions of suchan analysis.

Let us first consider resonators with inversion symmetry, namely the BC-SRR orthe NB-SRR. As both rings forming the structure are identical, it is expected that SRRresonances will appear as a consequence of the splitting of the resonances of a singlering. Therefore, the number of resonances in a given frequency interval for theNB- and the BC-SRR should be approximately twice the number of resonances ofits constitutive single rings inside such an interval. The analysis of Section 2.3.1(specifically, the analysis of the current distribution shown in Fig. 2.5c) can beeasily extended to the quasistatic resonances of the BC- or the NB-SRR of Figures2.7 and 2.9, respectively. From such an analysis it follows that the current distributionof these resonances must be antisymmetric. That is, the surface current on a givenpoint, (x, y), of a ring, namely ring A, Js,A(x, y), has the same amplitude but oppositesign as the surface current on the other ring, namely ring B, at the opposite point:Js,A(x, y) ¼ �Js,B(�x,�y). In fact, this property is directly related to the inversionsymmetry of NB- and BC-SRRs. Therefore, it can be extended to higher-order reson-ances, which can be classified as symmetric (with Js,A(x, y) ¼ Js,B(�x,�y)) andantisymmetric (with Js,A(x, y) ¼ �Js,B(�x,�y)). Thus, it is expected that eachsingle-ring resonance will split into two NB- or BC-SRR resonances, one symmetricand the other anti-symmetric. This behavior is illustrated in Figure 2.15, where thesplitting of the single-ring resonances in the symmetric and anti-symmetric NB-SRR resonances is shown. It follows directly from this classification (and from thedefinitions of the electric and magnetic dipolar moments) that the symmetric reson-ances cannot produce any magnetic dipolar moment, whereas the antisymmetric ones(which have a symmetric charge distribution) cannot produce any electric dipolarmoment. A particular case of this last property is demonstrated by the well-knownbehavior of the first (quasistatic) antisymmetric resonance for the considered NB-and BC-SRRs; it is essentially a magnetic resonance, which does not presentcross-polarization effects (see Section 2.3.2). From the above schema, it followsthat the second NB- and BC-SRR resonances are symmetric and electric (there isno magnetic dipolar moment associated with such resonance).

The onset of the resonances for the conventional EC-SRR is quite different.Because the rings forming the particle are not identical, its individual frequencies


of resonance must be different. When they are strongly coupled to form an EC-SRR,the frequencies of resonance of each ring will change as a consequence of the coup-ling. Thus, the final number of these resonances in a given frequency interval willremain constant; that is, it will be roughly the same as for the system of twodecoupled rings. The stronger the coupling, the wider the difference between theEC-SRR resonances and those of its constitutive rings. From the above consider-ations, it directly follows that the classification of symmetric and antisymmetricdoes not hold for the EC-SRR resonances. Instead, as is illustrated in Figure 2.16,higher-order EC-SRR resonances resemble more the individual resonances of theindividual rings forming the structure. Thus, bianisotropic effects can be present,in principle, in all resonances. However, it has been experimentally found that thesecond resonance of the EC-SRR does not present cross-polarization effects [30].That is, it has only an associated electric dipolar moment, without any detectableadditional magnetic dipolar moment. We do not have, for the moment, any theoreticalexplanation for this remarkable fact.

As has already been mentioned, higher-order SRR resonances may have effects onthe behavior of SRR-based metamaterials. In particular, the second resonance maystrongly affect the behavior of such metamaterials if the coupling between therings forming the SRRs is weak. In such a case, the first and the second resonancesappear closely spaced. Because the second resonance produces a strong electricdipole, a dielectric permittivity higher than expected may appear around such aresonance. This region of unexpectedly high electric permittivity may overlap with

FIGURE 2.15 Resonances of an NB-SRR. The resonances appear as dips in the trans-mission coefficient of a microstrip line coupled to the NB-SRR. The NB-SRR resonancesare labeled as “symmetric” (S) and “antisymmetric” (A) according to the classification inthe text. The resonances of one of the rings making the NB-SRR are also shown. The NB-SRR parameters are external radius rext ¼ 6:217mm, strip width c ¼ 0.56mm, width separ-ation d ¼ 0.37mm. The substrate parameters are thickness h ¼ 0.49mm, dielectric constant1 ¼ 2.4310. (Source: Reprinted with permission from [30]; copyright 2005, AmericanInstitute of Physics.)


the region of negative magnetic permeability, above the first SRR resonance.Conversely, if both SRR rings are strongly coupled, the second SRR resonancewill appear far away from the first one. It will then produce a frequency band ofhigh positive/negative dielectric permittivity neatly separated from the negative per-meability frequency band. This effect can be useful for some applications [31].

2.3.5 Isotropic SRRs

Small magnetic resonators showing a purely magnetic and isotropic response are ofgreat interest for the design of isotropic magnetic metamaterials. It is clear that themagnetic response of any planar SRR, as those analyzed in the previous sections,must be highly anisotropic. In addition, depending on its specific topology, anSRR may also be bianisotropic; that is, it can show both magnetic and electricresponse at the same resonance. The first attempt to obtain an isotropic SRRdesign was reported in [32]. In this work, two SRRs of cylindrical shape wereattached in order to obtain a composite SRR configuration, which was shown to beisotropic in two dimensions. Unfortunately, it was not possible to generalize thisapproach to obtain an isotropic three-dimensional configuration. A different approachto the same problem was reported in [33] and [34]. In these works, cubic arrays ofSRRs and Omega resonators [35] were proposed in order to obtain a composite iso-tropic magnetic resonator. However, it has been shown [36] that couplings betweenadjacent elements can destroy the isotropic response of such arrangements.

A systematic approach to the design of isotropic SRR composites must start fromthe group theory of symmetry transformations in periodic systems [36]. It will be

FIGURE 2.16 Resonances of an EC-SRR determined as in Figure 2.15. The EC-SRR andsubstrate parameters are the same as in Figure 2.15. Only the first and second resonancescan be identified as approximately symmetric or anti-symmetric. Higher-order resonancesare similar to those of the outer (O) or inner (I) isolated rings. (Source: Reprinted with per-mission from [30]; copyright 2005, American Institute of Physics.)


assumed, as usual, that the electromagnetic response of the composite is accuratelydescribed by some linear polarizabilities, which form some second-order tensors orpseudotensors. It is well known [21] that there are 32 symmetry groups for periodicalsystems, which can be classified in seven systems. It is also well known that only thesymmetry groups of the cubic system reduce the second-order tensors to scalars, thusimposing an isotropic behavior. Moreover, if cross-polarizabilities must be forbidden,inversion symmetry must be included in such groups. This reduces the possible sym-metry groups to only two: the Oh and Th groups in Schonflies notation [36]. The Oh

group includes all the symmetry rotations of the cube, and the inversion. It has thehighest symmetry, including 48 symmetry operations. In [37], a resonant magneticstructure showing this symmetry was proposed for magnetic metamaterial design.9

The Th group is generated by the rotations of symmetry of the tetrahedron and theinversion. It has 24 operations of symmetry, thus imposing lower symmetry con-straints than the Oh group. This fact results in potentially simpler designs. As anexample, two isotropic magnetic resonators pertaining to the Th symmetry groupare shown in Figure 2.17 [36]. The structure shown in Figure 2.17a is a modificationof the SRR composite previously proposed in [32]. It is the composition of three 2-SRRs (see Section 2.3.2) of cylindrical shape. This last structure shows two orthog-onal symmetry planes (the x ¼ 0 and the y ¼ 0 planes in Fig. 2.10), which behave asvirtual perfectly conducting planes at resonance. Therefore, the inclusion of twoadditional 2-SRRs at these planes (as in Fig. 2.17a) will not affect the behavior ofthe former 2-SRR when it is illuminated by an orthogonal external magnetic fieldBext ¼ Bextz. Thus, by superposition, the whole structure must exhibit an isotropicand purely magnetic response. The frequency of resonance of the composite mustbe the same as for each individual 2-SRR (2.54), which is twice that of an EC-SRR of the same size and characteristics. However, this is not a big problem inthat the broadside coupling between the rings allows for very high interringcapacitances.

FIGURE 2.17 Two isotropic composite SRR resonators belonging to the Th symmetrygroup. Both resonators show a purely magnetic response at the first resonance. (Source:Reprinted with permission from [36]; copyright 2006, American Institute of Physics.)

9In fact, for left-handed metamaterial design, since the proposed structure also included a regular arrayof wires.


An other example of an isotropic magnetic resonator belonging to the Th group ofsymmetry is the structure shown in Figure 2.17b. This is a composite of six two-splitBC-SRRs (the structure shown in Fig. 2.7, with an additional symmetric cut on eachring). This composite also belongs to the Th symmetry group. Thus, it will also showan isotropic and purely magnetic response [36].

2.3.6 Scaling Down SRRs to Infrared and Optical Frequencies

After the demonstration of negative permeability in the microwave range [4], therehas been a continuous effort for pushing up the frequency of operation of magneticmetamaterials to terahertz and optical frequencies. A rather obvious strategy in thisregard is to take advantage of the linearity of electromagnetism and scaling downthe size of the well-known metallic SRR, so that its frequency of operation ispushed up [38,39]. However, the application of this procedure has a limit, because,when frequency increases, metals cannot be characterized as near-perfect conductors,as was assumed in Section 2.3.1. From terahertz to ultraviolet frequencies, mostmetals can be characterized by the well-known Drude complex permittivity,

1 ¼ 10 1�v2p

v(v� jfc)

!, (2:78)

where vp is the plasma frequency, and fc is the collision frequency of the electrons.The effects of this new metal characterization on the SRR frequency of resonance andpolarizability will be considered throughout this section.

Before analyzing the behavior of an SRR, we will consider the more simple case ofa single metallic ring (Fig. 2.18a). We will consider, for simplicity, a metallic ring ofmean radius r, made with a cylindrical wire of diameter d and cross-sectionS ¼ p(d=2)2. For good conductors fp ¼ vp=2p is in the ultraviolet range (several

FIGURE 2.18 (a) Closed metallic ring and (b) the simplest split-ring configuration withinversion symmetry.


thousands of terahertz), and fc is between 10 THz and 100 THz. Therefore, roughlyspeaking, vp � 102fc: At infrared and optical frequencies, v is still smaller thanvp, say v . 0:1vp. Therefore, at such frequencies and below, j1j � 10, and equation(2.78) can be approximated as

1 ’ �10v2p

v(v� jfc): (2:79)

From the continuity of the normal component of jv1E at the metal–air interface, itdirectly follows that n � Er � 0, where Er is the electric field inside the ring, and nis the unit vector normal to the ring boundary. Thus, the electric lines of force arestrongly confined inside the wire, forming closed loops along the ring. Therefore,it makes sense to define a total current inside the ring as

It ¼ jv1Er S, (2:80)

where Er is the magnitude of the electric field inside the ring. This current includesboth ohmic and displacement currents, and is approximately uniform along thering due to the aforementioned confinement effect. Therefore, it is still possible todefine the ring magnetic inductance, Lm, as usual, that is,

Lm ;F(It)It

, (2:81)

where F(It) is the magnetic flux across the ring. Because the magnetic flux is relatedto the total current through Ampere’s law, Lm is still approximately given by equation(2.16). In fact, a small correction must be included in equation (2.16) in order to takeinto account the magnetic energy stored inside the wire, which cannot be neglectedfor nonperfect conductors. Assuming a uniform distribution of the current insidethe ring, the final expression for Lm is [13]

Lm ¼ m0r ln16dr

� �� 74

� �: (2:82)

Taking into account equations (2.79) and (2.80), the electromotive force around thering E ¼

ÞE dl ’ 2prEr can be written as

E ’ (Rþ jvLk)It, (2:83)


where R is the resistance of the ring

R ¼ 2prfcSv2

p10, (2:84)

and Lk is a magnitude with dimensions of inductance given by

Lk ¼2prSv2

p10: (2:85)

This inductance can be interpreted as being due to the kinetic energy of the electrons[40]. Putting all this together, the final equation for the current It on the ring isgiven by

{Rþ jv(Lm þ Lk)}It ¼ �jvFext, (2:86)

where Fext is the external magnetic flux across the ring. Therefore, as long asj1j � 10 (which is satisfied if v . 0:1vp ), the only effect of scaling down the ringis the onset of the additional inductance Lk. From equations (2.82) and (2.85) itfollows that [41]

LmLk

� 2pS

l2p, (2:87)

where lp ¼ 2pc=vp is the “plasma wavelength”, that is, the free-space wavelength atthe plasma frequency.

Let us now consider what happens when some capacitive gaps are added to theclosed ring of Figure 2.18, in order to obtain a resonant behavior. The simplest res-onant configuration having inversion symmetry (in order to avoid cross-polarizationeffects) is shown in Figure 2.18b.10 However, the present analysis is not restricted tosuch a configuration, and more complicated geometries can be considered, includingthe SRRs reported in Sections 2.3.1 and 2.3.2. We will assume, as in the precedingparagraph, that j1j � 10, so that equation (2.79) holds. If the gaps are sufficientlynarrow, the electric field lines of force are still confined inside the ring, passingacross the gaps from one half ring to the other. Moreover, from the continuity ofthe normal component of jv1E at the gap boundaries, it directly follows that thetotal current It is still uniform along the ring and gaps.11 Also, because the

10This configuration was proposed and analyzed in [40].11This behavior is quite similar to the behavior of the magnetic flux through the arms and between the polesof an electromagnet, as described in many elementary textbooks of electricity and magnetism (see, e.g.,[42]).


permittivity inside the metal is essentially real and negative, the continuity of 1Eimplies that the sign of the electric field lines is reversed in the gaps, as is shownin Figure 2.19.

Provided the aforementioned hypotheses are fulfilled, the equation for the totalcurrent in the resonator of Figure 2.18b is gives as

Rþ jv(Lm þ Lk)þ1jvC

� �It ¼ �jvFext, (2:88)

where C is the total capacitance provided by the series connection of the gaps (i.e.,C ¼ Cg=2, where Cg is the capacitance of each gap). From equation (2.88), thetotal current, as well as the associated magnetic moment m ¼ pr2It, can be obtained.Finally, the magnetic polarizability of the ring is obtained as

ammzz ¼ a0

v2

v20 � v2 þ jvg

� �and a0 ¼

p2r4

Lm þ Lk, (2:89)

where v0 is the frequency of resonance,

v0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1(Lm þ Lk)C

r, (2:90)

and

g ¼ R

Lm þ Lk: (2:91)

Such expressions are, in fact, quite similar to equation (2.45). The only differencewith equation (2.45) is the presence of the kinetic inductance Lk. Therefore, wecan conclude that the analysis developed throughout the previous sections is still

FIGURE 2.19 Schematic illustration of the behavior of the electric field lines inside the reso-nator of Figure 2.18b.


valid at optical and infrared frequencies, provided the presence of the kinetic induc-tance (2.85) is taken into account. The main restriction for the validity of such ananalysis is that the condition j1j � 10 must hold at the considered frequency.

Let us now consider the behavior of the SRR polarizability when the resonator isscaled down in order to increase its frequency of resonance. Because, according toequation (2.85), Lk scales as 1/r, for small SRR sizes it dominates over Lm, whichscales as r. Therefore, for sufficiently small ring sizes, Lk � Lm, and the frequencyof resonance (2.90) saturates [40] to the value

v0 ¼ffiffiffiffiffiffiffiffi1

LkC

r(2:92)

which does not depend on SRR size. Also the loss factor saturates to

gs ¼R

Lk¼ fc: (2:93)

Finally, the amplitude of the magnetic polarizability scales down as

a0 !p2r4

Lk¼ v2

p10r3S: (2:94)

Let us consider, as in previous paragraphs, a cubic array of SRRs of lattice constant a.As was discussed in Section 2.3.3, the magnetic susceptibility xzz can be approxi-mated as xzz � m0a

mmzz =a3. Thus, from the previous considerations, it follows that,

for sufficiently small SRR sizes, this susceptibility behaves as

xzz � x0v2

v2s � v2 þ jvgs

� �and x0 ¼ m0

a0

a3� (2p)2

r

a

�3 S

l2p: (2:95)

This expression shows that, when the structure is scaled down in the optical range, theamplitude of the magnetic susceptibility also scales down as the square of SRR size(see also Problem 2.11).12 The above discussion shows that scaling down SRR mediain order to achieve a negative magnetic permeability at infrared and optical frequen-cies has two main limitations: saturation of frequency of resonance and decrease ofthe magnetic response. Such effects appear when the key parameter S=l2p becomessmall, which, according to equation (2.87), makes the kinetic inductance dominantover the magnetic one.

Recently, kinetic inductance of conductors has been invoked in order to developnano-inductors at optical frequencies [43]. According to our previous discussion, a

12In the RF and microwave range, however, because Lm scales down as r, the amplitude x0 does not varysubstantially when the structure is scaled down.


metallic wire of length l and section S has a kinetic inductance that is given by a directgeneralization of equation (2.85); that is,

Lk ¼l

Sv2p10

: (2:96)

Therefore, it can be considered as a nanoinductor with the above inductance. In fact,any nanocircuit is composed of some closed loops of current, and in order to applyKirchoff’s laws, the magnetic inductance of every loop should be much smaller thanthe inductance of each inductor in the circuit. For instance, the SRR of Figure 2.18bcan be considered a nanocircuit formed by two nanoinductors (the wires) and twonanocapacitors (the gaps), provided its magnetic inductance Lm is much smallerthan the kinetic inductance Lk. From equation (2.87) it follows that this conditionis satisfied if

S=l2p � 1: (2:97)

In fact, it can be easily realized that the same condition must hold for any closed loopof any hypothetical nanocircuit. Therefore, equation (2.97) seems to be a necessarycondition for the applicability of the nanoinductor concept.

Higher-order SRR resonances in the optical range are dynamic plasmonic reson-ances. Closed metallic rings made of metallic wires resonate when the ring perimeteris equal to a wavelength of one of the plasmons that can be excited along the wire[44]. From this result it can be guessed that n-cuts SRRs will resonate when eachpiece of wire forming the SRR is a multiple of a half-wavelength of a plasmon pro-pagating along the wire. Such resonances can also generate electric and magneticdipoles, so their analysis (either to avoid, or to take advantage of them) may alsobe important for SRR characterization at infrared and optical frequencies. Such analy-sis is, however, beyond the scope of this book.

2.4 SRR-BASED LEFT-HANDED METAMATERIALS

The first artificial left-handed medium, designed by David Smith and co-workers [4],came only one year after Pendry’s proposal for SRR-based artificial negative mag-netic permeability media [3]. Smith’s first design was a combination of wires(which provide the negative permittivity) and SRRs (which provide the negative per-meability) [4]. Soon after, negative refraction in left-handed media was experimen-tally demonstrated [45], using a very similar design. Since then, wires and/ormetallic plates assembled with SRRs have been the most popular combination forleft-handed metamaterial design. Through this section, the main characteristics ofthese proposals will be described in detail. Other proposals will be analyzed inSection 2.5.


2.4.1 One-Dimensional SRR-Based Left-Handed Metamaterials

The first practical realization of a left-handed metamaterial [4] was, in fact, a one-dimensional realization. The reported structure is shown in Figure 2.20. It consistsof an array of metallic wires and EC-SRRs placed between two parallel metallicplates. The structure was illuminated by a plane wave polarized with the electricfield parallel to the wires, and the magnetic field perpendicular to the EC-SRRs, sothat the parallel metallic plates simulate an infinite periodic medium along the wireaxis. In a previous experiment, propagation through a structure as in Figure 2.20,but without the wires, was checked. A stopband was observed in this last structurenear the frequency of resonance of the SRRs (solid line in Fig. 2.21). This stopbandis consistent with the negative magnetic permeability associated with the SRRs.When wires were introduced, the aforementioned stopband switched to a passband(dashed lines in Fig. 2.21). Because the frequency of the experiment was wellbelow the plasma frequency of the system of wires, the aforementioned passbandis consistent with the combination of the negative permittivity of the wire system(see Section 2.2.2) and the negative permeability of the SRR system. Therefore, itwas the first experimental observation of electromagnetic propagation through aleft-handed medium. Backward-wave propagation was tested by computing the dis-persion relation by electromagnetic simulation. The results of such simulations, aswell as the dispersion relations obtained from the theory reported in this chapter(see Sections 2.2.2 and 2.3.3), are shown in Figure 2.22. The slight mismatchbetween the negative-m stopband and the left-handed passband comes from thebianisotropy of the EC-SRR system [17] (see Problem 2.8).

The structure of Figure 2.20 only shows a left-handed behavior for a specific polar-ization and direction of propagation of the electromagnetic field. Therefore, it can beconsidered a one-dimensional metamaterial, showing negative effective permittivityalong the wires, and—neglecting bianisotropy—negative permeability along the SRR

FIGURE 2.20 Sketch of Smith’s experiment (top view). The relative location of wires andSRRs is illustrated. The whole system is placed between two (upper and bottom) metallicplates, and simulates an infinite medium along the wire axis. The orientation of the EC-SRRs with regard to the x–y–z axes is the same as in Figure 2.12.

2.4 SRR-BASED LEFT-HANDED METAMATERIALS 81

axis. If the wire rows are substituted with metallic plates, a more simple one-dimensional structure is obtained [46]. Such a structure is shown in Figure 2.23. Itis made of a single row of SRRs placed inside a cutoff square waveguide, along itsmiddle E-plane. In such a configuration, the effective negative permittivity (2.2) isprovided by the cutoff waveguide and the effective negative permeability by theSRRs: meff ¼ m0(1þ xm,zz), with xm,zz given by equation (2.74). The left-handedpassband measured in the device of Figure 2.23 is shown in Figure 2.24.Backward-wave propagation along this device has been experimentally demonstratedin [48]. As in the structure reported in [4], bianisotropic effects are also present in thestructure of Figure 2.23. These effects can be eliminated if BC-SRRs are used insteadof EC-SRRs [50]. This last structure provides an easy way of simulating propagationin left-handed metamaterials. It will be extensively used in Section 2.4.5 to check theaccuracy of the models proposed in this chapter for SRR-based metamaterials.

It will be illustrative to mention here that the results reported in Figures 2.23 and2.24 can also be interpreted in terms of a cutoff waveguide filled by an uniaxialmedium with negative permeability along the SRR axis [47,48]. This interpretationmodels the structure of Figure 2.23 as a waveguide filled by a uniaxial anisotropicmagnetic medium with mxx ¼ myy ¼ m0 and mzz obtained from equation (2.74).13

From Maxwell is equations, it is deduced that the dispersion relation for such

FIGURE 2.21 Transmission coefficient through the structure proposed in [4] (see Fig. 2.20)when only SRRs are present (solid line), and when wires and SRRs are present (dashed line).(Source: Reprinted with permission from [4]; copyright 2000, by the American PhysicalSociety.)

13Bianisotropy, as well as the small correction to 1yy due to the nonresonant electric polarizability of theSRRs, are neglected in this analysis. However, they can be easily included in the model.


a waveguide is given by

k2 ¼ v2mzz10 1� v2c

v2

� �¼ v21effmeff , (2:98)

where vc is the cutoff frequency of the hollow waveguide. This dispersion relation isexactly the same dispersion relation that is deduced from the analysis in the previousparagraph. Therefore, both models are fully equivalent, leading to the same dis-persion relation. In our opinion this fact does not mean that the interpretation pre-viously developed in this section is erroneous.14 It only shows that, in order toobtain a left-handed behavior by loading a waveguide below cutoff with magneticresonators, such resonators must interact with the transverse components of the mag-netic field and not with the longitudinal components of such a field. This behavior canbe obtained from SRRs, or from a hypothetical uniaxial magnetic medium.15

FIGURE 2.22 Phase advance (ka) as a function of frequency for the SRR medium ofFigure 2.20 without wires (dotted lines), and phase advance for the SRR and wire left-handedmedium of Figure 2.20 (solid line). The experimental (see Fig. 2.21) left-handed passbandDv, aswell as the experimental stopbandDvþ dv, are also shown in the figure. The SRRdimen-sions are c ¼ 0.8 mm, d ¼ 0.2 mm, rint ¼ r2 d/2 2 c ¼ 1.5 mm, and t ¼ 0.216 mm. The rela-tive dielectric permittivity of the circuit board is 1r ¼ 3:4, and the lattice constant a ¼ 8.0 mm.(Source: Reprinted with permission from [19]; copyright 2003, IEEE.)

14See author’s reply to [47] for a deeper discussion on this point.15There are at least two more interesting interpretations of the results reported in [46]. The first was pro-posed by Belov [49], and considers the SRR-loaded waveguide as equivalent to a system of magnetic scat-terers (the SRRs) propagating a wave that is stationary in the z-direction (with kz ¼+p/a, where a is thewidth of the waveguide). Another possible interpretation considers the results reported in [46] as the com-position of two magnetostatic waves in an infinite SRR medium (see Problem 2.10) with kz ¼+p/a, ky¼0 and kx given by equation (2.167) (see Problem 2.10). It can be easily shown that for p/a � k0 ;vffiffiffiffiffiffiffiffiffiffiffi10 m0

p, the value of kx obtained from equation (2.167) coincides with equation (2.98).


From the dispersion relation (2.98)—whatever the physical interpretation of suchan equation—it is deduced that, if the hollow waveguide is above cutoff, the left-handed passband associated with the negative magnetic permeability becomes a stop-band. This effect has been shown experimentally in [51]. In such an experiment, twohollow waveguides—one above and the other below cutoff—were loaded by equis-paced BC-SRR rows. The resulting transmission is shown in Figure 2.25, where it canbe clearly seen how the left-handed passband of the narrow waveguide becomes astopband in the wider waveguide. The mismatch between both bands, near the BC-SRR resonance, can be explained by the high losses and anomalous dispersion atsuch frequencies. The behavior shown in Figure 2.25 can be considered as thecounterpart, in waveguide configurations, of the passbands and stopbands ofFigure 2.21.

FIGURE 2.24 Transmission coefficient through the structure of Figure 2.23. (Source:Reprinted with permission from [46]; copyright 2002, American Physical Society.)

FIGURE 2.23 Sketch of the one-dimensional left-handed medium simulation experiment in[46]. [Source: Reprinted with permission from [46]; copyright 2002, American PhysicalSociety.)


2.4.2 Two-Dimensional and Three-Dimensional SRR-BasedLeft-Handed Metamaterials

The structures reported in the previous section exhibit a left-handed behavior for aspecific direction of propagation, and for a specific polarization of the electromag-netic wave. Therefore, they are not suitable for the demonstration of negative refrac-tion at the interface with an ordinary medium. For this purpose, a two-dimensionalisotropic left-handed medium is necessary. Such a medium was designed and man-ufactured soon after the first demonstration of a left-handed medium [52]. It was amodification of the earlier design of [4], consisting of an orthogonal arrangementof dielectric circuit boards with EC-SRRs and metallic strips printed on each side,as is shown in Figure 2.26. The whole array is placed between two horizontal metallicplates connected to the strips, so that an infinite structure is simulated along the z-axis.As in the structure of Figure 2.20, the negative permittivity comes from the metallicstrips, and the negative permeability from the SRRs. However, the whole structure is

FIGURE 2.25 Transmission coefficient through the BC-SRR loaded waveguides sketchedin the figure. The narrow waveguide cross-section was 6 6 mm2, and the wide waveguidecross-section was 30 6 mm2. The periodicity is a ¼ 6 mm. The BC-SRR parameters arerext ¼ 2.28 and c ¼ 0.5mm. The substrate parameters are 1 ¼ 2:4310 and thickness t ¼0.49mm. The metallization was in copper with a thickness h ¼ 35mm. (Source: Reprintedwith permission from [51]; copyright 2003, IEEE.)


now isotropic in two dimensions.16 Using the reported structure, negative refraction atthe interface between ordinary and left-handed media was first demonstrated [45]. It isinstructive to note that, in Figure 2.26, metallic strips are placed along the symmetryplanes of the SRRs. Such planes behave as virtually perfect electric conductors atresonance, a fact that minimizes the strip–SRR interaction (we will return to thispoint later, in Section 2.4.4).

An alternative design for two-dimensional isotropic left-handed metamaterials wasproposed in [50]. It consists of metallic plates and BC-SRRs arranged and illuminatedas is shown in Figure 2.27. As in previous designs, the negative permeability comesfrom the SRRs. However, in the present design, negative permeability comes from theparallel metallic plates (see Section 2.2.1). Because its fabrication only involves asingle printed circuit board, this design is more simple to manufacture than the pre-vious one. Moreover, the use of BC-SRRs ensures that cross-polarization effects arenot present along the structure. Negative refraction using the two-dimensional left-handed structure of Figure 2.27 was experimentally demonstrated in [53].

Compared with two-dimensional structures, the development of practical isotropicleft-handed media in three dimensions is still a challenging issue. In principle, a three-dimensional cubic array of connected wires could provide negative permittivity (seeSection 2.2.2), whereas magnetic permeability could come from a cubic arrangement

FIGURE 2.26 The basic unit of the two-dimensional left-handed metamaterial reported in[52] (left). The metallic strips (gray) and the EC-SRRs (black) are printed on opposite sidesof the dielectric substrate. The metamaterial was made from a two-dimensional periodicarray along the x- and y-directions of such basic units (right). The whole system was placedbetween two conducting plates parallel to the x–y plane that simulates an infinite medium inthe z-direction.

16In fact, regarding the analysis in Section 2.3.3, this structure must be biisotropic in two dimensions (seeProblem 2.9). However, for the structure proposed in [52] this effect is small except near the resonance, andcan be neglected in a first-order approximation.


of any of the isotropic SRRs reported in Section 2.3.5. However, practical technologi-cal limitations to this procedure may be enormous. First, isotropic SRRs are rathersophisticated structures, much more difficult to manufacture than planar SRRs.Then, the number of SRRs involved in a three-dimensional design grows as l3,where l is the typical length of the structure. Finally, to assembly all the elementsis not a trivial task. Moreover, once backward-wave propagation and negative refrac-tion in left-handed media has been demonstrated using one- and two-dimensionalstructures, the scientific interest in three-dimensional structures decays. However,these designs may still be of interest for many practical applications, such as three-dimensional lenses or coats. In [37] an isotropic three-dimensional left-handed meta-material design was proposed and validated by numerical simulations. Other similardesigns, using a cubic network of metallic wires in combination with different isotro-pic SRRs, such as those shown in Figure 2.17, can be envisaged.

2.4.3 On the Application of the Continuous-Medium Approach toDiscrete SRR-Based Left-Handed Metamaterials

The continuous-medium approach for the description of electromagnetic properties ofa discrete medium always involves an averaging—or homogenization—procedure.This homogenization only has sense if the variation of the average field is small atthe scale of the constitutive elements of the medium. This condition is usuallyexpressed as follows: The size of the elements must be small as compared with thewavelength of the host medium, lh, at the frequency of operation. Stated in this

FIGURE 2.27 Sketch (front view) of the two-dimensional isotropic left-handed meta-material proposed in [50]. A square array of BC-SRRs is printed on a dielectric board,placed between two parallel metallic plates.


form, the above condition is approximately satisfied by the reported SRR-based meta-materials. In fact, in all the experiments reported along the previous section, the unitcell size was �lh=10. However, the aforementioned condition cannot be acceptedwithout discussion. All left-handed metamaterials analyzed in this section are res-onant structures, whose wavevector varies from 0 to 1 in a small frequency band(see, e.g., Fig. 2.22). Therefore, the internal wavelength of the metamaterialli ¼ 2p=k varies from very large to very small values in the operation band.Because it is the internal wavelength that determines the scale of variations of the aver-aged field, we conclude that the size of the metamaterial elements must be muchsmaller than li in order to justify the homogenization. It is apparent that this conditioncannot be fulfilled in the whole left-handed passband. In particular (Fig. 2.22), itcannot be fulfilled at the lower frequencies of this passband, where li ! 0 (in fact,all the values of the phase advance outside the first Brillouin zone, i.e., higher thanp, have no physical meaning—they were only computed in order to determine thepassband predicted by the model). Therefore, for the lowest frequencies of the pass-band the discrete nature of the metamaterial must be taken into account. That is, themetamaterial should be explicitly analyzed as a three-dimensional periodic structure.Of course, the smaller the electrical size of the elements is (in terms of the host mediumwavelength), the smaller is such a frequency region.

Conversely, at the upper frequencies inside the passband, li ! 1, and thehomogenization makes sense regardless of the size of the elements. Therefore, itcan be concluded that the continuous medium approach in left-handed metamaterialsis valid at the high-frequency region of the passband, and becomes more and moreinaccurate as the frequency approaches the lower limit of the passband.

2.4.4 The Superposition Hypothesis

There is another important issue that has to be discussed in relation to the accuracy ofthe continuous-medium approach, as it has been applied in most of the reported con-tributions to left-handed metamaterials. In fact, all left-handed metamaterialsdescribed in this section (and in all the reported references), are the superpositionof two systems, one of them exhibiting negative permittivity and the othershowing negative permeability. It was assumed without discussion that the electricand magnetic susceptibilities of the resulting media are the superposition of the elec-tric and magnetic susceptibilities of the aforementioned isolated systems. Because theelements of both systems are placed very close in the metamaterial, and therefore theycan interact, this assumption is not apparent at all.

This topic was discussed in [54], where it was shown that a system of parallel wiresplaced in a host medium with negative permeability cannot propagate electromag-netic waves. This result could be expected from the expression for the inductanceof the wires (2.6): When the permeability of the surrounding medium is negative,such inductance also becomes negative, thus destroying the negative permittivity pro-vided by the wires. From such a result, it was concluded in [54] that the superpositionof a system of wires and a system of SRRs cannot be seen, in general, as a mediumwith simultaneously negative permittivity and permeability. The argument in [54]


becomes even more apparent if we consider the structure shown in Figure 2.23 (or inFig. 2.27). It can be easily shown [47] that electromagnetic waves cannot propagatealong a waveguide filled by an isotropic medium of negative permeability.

The meaning and validity of the superposition hypothesis for left-handed mediabased on SRRs and wires or plates was further discussed in [55] and in theauthor’s reply to [47]. It is important to recall that the continuous-medium approachonly accounts for the behavior of the average field, and not for the field behavior at themuch smaller scale of the metamaterial elements. At this scale, the short-rangequasielectrostatic and quasimagnetostatic interactions between the metamaterialelements determine the coupling between them. Therefore, the relative dispositionof such elements is crucial for the accuracy of the aforementioned superpositionhypothesis. Systems providing negative permittivity and negative permeabilityshould be placed in such a way that the interaction between its elements throughits quasistatic fields is minimized. This is actually the case in Figure 2.20, wherethe magnetic flux created by a wire on its nearest SRRs vanishes, so that the wire’sinductance is not affected by the SRRs. This is also the case in Figure 2.26, wherethe metallic strips are placed in the plane of symmetry of the SRRs, which behaveas a virtual perfect conductor (see Section 2.4.2). In fact, it has been shown that ifthe SRRs of Figure 2.20 are placed along the wire rows (where magnetic interactionhas a maximum), and not between these rows, as in the actual experiment, the left-handed behavior disappears [56] (both configurations are illustrated in Fig. 2.28).

Regarding the structures where negative permittivity comes from metallic wave-guides or plates—as in Figures 2.23 and 2.27—in order to obtain a left-handed beha-vior, the SRRs must interact with the transverse magnetic field, but not with thelongitudinal magnetic field of the structure. Note that the transverse component ofthe magnetic field, after averaging, produce the macroscopic magnetic componentof the left-handed wave. However, due to the symmetry of such devices, theaverage longitudinal magnetic field component vanishes. That is, in order toachieve a left-handed behavior for the average wave, the magnetic resonators mustinteract with the macroscopic field components, and not with the microscopic ones.

FIGURE 2.28 (a) Sketch (top view) of the configuration of Figure 2.20, where a left-handedbehavior was observed at some frequency range [4]. (b) Sketch (top view) of a configuration inwhich there is no wave propagation at the same frequencies [56].


In summary, it can be concluded that, in general, a superposition of two systems,one of them providing negative permittivity and the other providing negative per-meability, does not make a left-handed medium. However, there are some clever rela-tive dispositions of the elements of both systems for which the aforementionedsuperposition hypothesis is fulfilled, and the left-handed behavior is achieved.Ultimately, it is the agreement with the experiments that provides the final proof ofthe superposition hypothesis for a given design.

2.4.5 On the Numerical Accuracy of the Developed Model forSRR-Based Metamaterials

The numerical accuracy of the reported theory for the computation of the constitutiveparameters of left-handed metamaterials has already been illustrated in Figure 2.22,where it was comparedwith simulations and experimental results. It has also been exper-imentally tested in [51]. In this paper a BC-SRR loaded cutoff waveguide, similar to thatshown in Figure 2.23, was used in order to experimentally check the theoretical model(the only difference with Fig. 2.23 was that BC-SRRs instead of EC-SRRs were used).Electromagnetic propagation through three similar devices of different lengths, with 10,20, and 30 consecutive BC-SRRs, was studied. In particular, the time delay and theattenuation were measured. From these results, the slow-wave factor c/vg (where vg isthe group velocity) and the attenuation constant of the metamaterial were extracted ina quite direct way (see [51] for more details). Theoretical and experimental results fora particular structure are shown in Figures 2.29 and 2.30. The slow-wave factor canbe directly related to the phase constant, k, through vg ¼ j@v=@kj. Therefore, itsmeasurement can be considered equivalent to a measurement of the dispersion relation

FIGURE 2.29 Theoretical and experimental values for the slow-wave factor along the narrowBC-SRR loaded waveguide of Figure 2.25. Waveguide and BC-SRR characteristics are as inFigure 2.25. (Source: Data reproduced with permission from [51]; copyright 2003, IEEE.)


v(k) of the metamaterial. On the other hand, the measurement of the phase and attenu-ation constants in the considered structure corresponds to the measurement of thesequantities on a typical SRR-based left-handed metamaterial (see Sections 2.4.1 and2.4.2). Therefore, such measurements can be considered to be a good test for theaccuracy of the model developed in this chapter. It is clear from the figures that themodel provides a good qualitative picture of the metamaterial behavior, reproducingthe passband width and location with only a few percent of error. Attenuation alsoseems to be approximately predicted. The main disagreement is in the slow-wavefactor, that is, in the slope of the dispersion relation v(k). This fact can be related toa drawback in the application of the continuous-medium approach to a discrete meta-material with a periodicity not smaller that l/10. In fact, as has already been discussedin Section 2.4.3, for the lower frequencies of the passband, the internal wavelengthbecomes very small, and the discrete nature of the metamaterial cannot be ignored.It can be guessed that for smaller periodicities this drawback could be overcome.

In summary, it can be concluded that the reported theory not only provides phys-ical insight on the qualitative behavior of left-handed SRR-based metamaterials, butalso reasonable quantitative results (only limited by general considerations about thevalidity of the homogenization procedures). This is a remarkable fact, because thedeveloped theory is analytical and, therefore, much more simple to implement thanstandard electromagnetic simulation techniques.

2.5 OTHER APPROACHES TO BULK METAMATERIAL DESIGN

There are two main approaches for metamaterial design at microwave frequencies.One of them was developed in the previous section. The other, introduced

FIGURE 2.30 Theoretical and experimental values for the attenuation factor along thenarrow BC-SRR loaded waveguide of Figure 2.25. (Source: Data reproduced with permissionfrom [51]; copyright 2003, IEEE.)

2.5 OTHER APPROACHES TO BULK METAMATERIAL DESIGN 91

simultaneously in [5] to [7], will be described in detail in the next chapter. Apart fromthese two main approaches, there are many other interesting proposals that can be alsoconsidered as bulk metamaterial designs (as they were defined at the beginning of thischapter). Some of them are described in this section.

2.5.1 Ferrite Metamaterials

There are two main groups of materials in nature showing negative permittivity orpermeability and small losses. One of them comprises semiconductors and metalsat infrared and optical frequencies. As has already been mentioned, metals and semi-conductors are accurately described by permittivity (2.78). At frequencies betweenthe collision frequency of the electrons fc and the plasma frequency fp, this dielectricconstant becomes approximately real and negative. For this reason, such materials aresometimes referred as solid-state plasmas. The other group corresponds to saturatedferrimagnetic materials (ferrites), which have long been used in microwave technol-ogy, and show negative magnetic permeability near the ferrimagnetic resonance.Unfortunately, both groups present these interesting properties at quite differentfrequency bands, which prevents their combination in designing left-handedmetamaterials.17

Before proceed with the description of ferrite-based left-handed metamaterials, wewill briefly describe the main characteristics of microwave propagation in ferrites.18

Ferrites, when magnetized to saturation by an external DC magnetic field H0, acquirea resonant anisotropic magnetic permeability. Low-loss cubic ferrites, such as yttriumiron garnets (YIG), are accurately described by the well-known Polder permeabilitytensor19 [58,59]

��m ¼ m0

��mt 00 mz

� �¼ m0

m jk 0�jk m 00 0 1

0@

1A, (2:99)

where the external magnetic field is assumed along the z-axis (H0 ¼ H0z) and

m ¼ 1þ vMvH

v2H � v2

(2:100)

and

k ¼ vvM

v2H � v2

, (2:101)

17Recently, the use of superconductors has been proposed to overcome this difficulty [57].18See, for a more detailed analysis, [58,59].19In such ferrites the only source of anisotropy is the DC fieldH0. Other ferrites with lower symmetry, suchas hexagonal ferrites, are described by a similar tensor, which takes into account the intrinsic anisotropy ofthe material.


where vH is the ferrimagnetic resonance frequency given by

vH ¼ gH0 (2:102)

and

vM ¼ 4pgMs, (2:103)

whereMs is the saturation magnetization of the ferrite, and g is the gyromagnetic ratiog ¼ 1:76 107 rad=sec Oe (c.g.s. units).20 It may be worth noting here that thePolder tensor (2.99) is not symmetric. Therefore, ferrites are nonreciprocal media,and many practical applications of ferrites in microwave technology come fromsuch nonreciprocity. In fact, the Polder tensor becomes transposed if the sign ofthe biasing magnetic field H0 is reversed. This fact can be expected from thegeneral properties of generalized susceptances in magnetized media [21]. A general-ized reciprocity theorem for ferrite media was given in [60].

It is clear from equation (2.99) that for v . vH, k becomes negative and that, forvH , v ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivH(vH þ vM)

p, m also becomes negative. Therefore, ferrites exhibit

negative permeability in a significant frequency range near the ferrimagnetic reson-ance. Regarding plane-wave propagation, the solution to the Maxwell equationswith the tensor permeability (2.99) gives four solutions for the wavevector. Forpropagation along the biasing field H0, such solutions are circularly polarizedplane waves. These waves have different phase constants for the different polariz-ations, given by

k+ ¼ +v

ffiffiffiffiffiffiffiffiffiffi1m+

eff

q, (2:104)

where 1 is the dielectric constant of the ferrite, and m+eff is an effective magnetic

permeability given by the two eigenvalues of ��mt,

m+eff ¼ m0(m+ k), (2:105)

where the “þ” sign stands for the right-handed circularly polarized (RCP) wave andthe “2” sign for the left-handed circularly polarized (LCP) wave21 with regard to thedirection of the biasing field H0. From equation (2.105), it is deduced that there is aregion of negative mþ

eff for

vH , v , vH þ vM: (2:106)

20The reported expression for the Polder tensor neglects losses. Expressions including losses may be foundin many textbooks, as, for instance, in [58] or [59].21Here, left- and right-handed refer only to the polarization, and do not imply any backward behavior.


In this frequency range, the RCP wave sees a negative effective permeability, andbecomes evanescent.

For plane waves propagating in a direction orthogonal to the biasing fieldH0, thereare two kind of solutions to Maxwell’s equations. One of them is an ordinary TEMplane wave polarized with the magnetic field parallel to H0 and k ¼ +v

ffiffiffiffiffiffiffiffi1m0

p. The

other is an extraordinary TE plane wave, with the electric field parallel to H0 and theH field elliptically polarized in the plane orthogonal toH0. The dispersion relation forthe extraordinary wave is

k ¼ +vffiffiffiffiffiffiffiffiffiffi1meff

p, (2:107)

where

meff ¼ m0m2 � k2

m: (2:108)

From equation (2.108), a negative effective permeability stopband is deduced forextraordinary waves in the frequency range


p, v , vH þ vM: (2:109)

In the neighborhoods of current sources, at distances much smaller than the wave-length, magnetic fields are quasimagnetostatic. Such quasimagnetostatic fields obeyAmpere’s law.22 Therefore, in source-free regions, they are irrotational, and can beexpressed as the gradient of a scalar magnetic potential cm:

H ¼ �rcm: (2:110)

Source-free solutions for such quasistatic magnetic potential with the permeabilitytensor (2.99) leads to the magnetostatic wave equation [59]

r � ��m ¼ �rcm ¼ m@2

@x2þ @2

@y2

� �þ @2

@z2

� �fm ¼ 0: (2:111)

It is interesting to note that the cross-term k does not appear in the magnetostatic waveequation.23 Therefore, for the quasimagnetostatic field, the ferrite appears as an uni-axial magnetic medium with m? ¼ m0m and mk ¼ m0. Such an effective medium

22In other words, electric displacement current can be neglected.23However, for finite systems, it usually appears in the boundary conditions.


shows a negative transverse permeability in the range

vH , v ,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivH(vH þ vM)

p: (2:112)

Among the many similarities between ferrites and solid-state plasmas is the exci-tation of magnetostatic surface waves (MSSWs), similar to surface plasmons. SuchMSSWs appears in ferrite–metal and ferrite–air interfaces, for external magnetiza-tion parallel to the interface (see Problems 2.12–2.14). MSSWs are unidirectionalwaves, which propagate only in the direction defined by the cross-product H0 n,where n is the outward normal to the ferrite. Depending on the boundary conditions,such MSSWs can also be excited in ferrite slabs, in all the frequency range


p, v , vH þ vM: (2:113)

In addition to MSSWs, ferrite slabs can also support magnetostatic volume waves(MSVWs), with a trigonometric field dependence inside the ferrite. The range ofexcitation of MSVWs coincides with the range of negative m (2.112), and for mag-netization parallel to the slab and propagation parallel to H0, they are backward-waves. Backward MSSWs can also appear for some specific boundary conditions.24

Like surface plasmons, magnetostatic waves are slow waves with k � k0 ¼ vffiffiffiffiffiffiffiffiffiffim010

p.

In such waves, energy is mainly associated with the magnetic field, which can beaccurately determined from the magnetostatic wave equation (2.111).

The above paragraphs show the large variety of phenomena that may appear inferrite media, including regions of effective negative permeability, and backward-wave propagation. This variety is even higher in waveguide configurations [61].All these properties strongly suggest the usefulness of ferrites for designing micro-structured metamaterials, with novel electromagnetic properties. Regarding left-handed metamaterials, there is an interesting precedent for the results reportedin [46] using ferrite-loaded waveguides. In a paper published in 1955 [62],Thompson reported backward-wave propagation in a below-cutoff waveguide,filled with a ferrite magnetized along the waveguide axis. The physical interpretationof this effect relies on the effective negative permeability (2.105) seen for right-handed circularly polarized (RCP) waves in the frequency range (2.106). This effec-tive negative permeability, when combined with the effective negative permittivity(2.2) provided by the below-cutoff waveguide, produces the typical left-handed beha-vior: backward-wave propagation below the cutoff frequency of the hollow wave-guide. Note that, according to equation (2.99), the permeability of the ferrite alongthe waveguide axis is undoubtedly positive (mz ¼ 1). Therefore, only the transverseeffective polarizability is negative inside the waveguide. Such simultaneously posi-tive/negative axial/transverse permeability explains, according to the discussion inSections 2.4.1 and 2.4.4, the observed effect. Therefore, although there was obviouslynot any reference to left-handed media in [62], the reported structure can be

24Specifically, when there is a metallic plate at some distance from the ferrite interface [58].


interpreted as a one-dimensional left-handed metamaterial. More recently, the sameanticutoff behavior has been demonstrated in ferrite-filled rectangular waveguideswith transverse magnetization [63]. In this case, the effective negative magneticpolarizability (for the appropriate wave polarization) is provided by equation (2.108).

The substitution by a ferrite of the SRRs in the first Smith’s proposal for left-handed metamaterial design (Fig. 2.20) has also been considered [64]. The proposedstructure is the same as in Figure 2.3, with the space between the wires filled by ahomogeneous ferrite, magnetized in the direction parallel to the wires. Accordingto the previous analysis, a plane wave incident in the plane perpendicular to thewires, and with the electric field polarized parallel to the wires (and to the externalmagnetization) will see an effective permeability given by equation (2.108). Thiseffective permeability is negative in the frequency range given in equation (2.109).Thus, when combined with the negative permittivity provided by the wire system,it will produce a left-handed metamaterial in a such frequency range, at least forthe considered field polarization. The proposal has the additional advantages of iso-tropy in the transverse plane, and tunability by a proper adjustment of the externalmagnetization. However, such a proposal presents an important drawback. It canbe easily realized that, for the considered excitation, the magnetic field in the directionparallel to the wires vanishes. Therefore, only the magnetic permeability in the planeperpendicular to the wires affects the field behavior. Assuming that this permeabilitycan be approximated by the scalar permeability (2.108), the behavior of the structurewill be identical to that of a system of wires immersed in a homogeneous mediumwith such permeability. It has also been shown [54] (see Section 2.4.4) that suchsystem does not behave as a left-handed metamaterial. To circumvent this drawback,the first proposal in [64] was modified, so that the wires were surrounded with adielectric cladding [65]. This cladding strongly affects the wire inductance (2.6),such that it remains positive.

As was mentioned at the beginning of this section, metals and semiconductorsbehave as solid-state plasmas, presenting a permittivity whose real part is negativeat frequencies below the plasma frequency (which is usually located between theinfrared and the ultraviolet frequency ranges). However, at microwave frequencies—where the ferrimagnetic resonance of known ferrites is located—the imaginary partof the permittivity of metals and semiconductors dominates over the real part. Thisfact seems to prevent the design of left-handed composites by taking advantage ofthe simultaneously negative real parts of the permeability and permittivity of ferritesand solid-state plasmas. Recently, the use of superconductors has been proposed toovercome this difficulty [57]. The proposed structure consists of a set of stacked fer-romagnetic25 and superconducting very thin layers. Magnetization is in the directionparallel to the layers. Normal incidence of plane waves polarized with the magneticfield perpendicular to the external magnetization is considered. For these waves, theferromagnetic layers present the effective permeability (2.108), which is negative inthe frequency range of expression (2.109). Superconducting layers are thinner than

25Results in [57] were reported for insulating ferromagnetic layers, with a behavior similar to those of fer-rimagnetic layers.


the penetration depth at the frequency of operation, and present a permittivity that ismostly real and negative. Thus, the resulting structure is quite similar to a stratifiedsystem of stacked 1-negative and m-negative layers. The properties of such asystem were analyzed in [66] (see also Chapter 1). For thin layers and normal inci-dence, it behaves as a one-dimensional left-handed medium, with a permeabilityand a permittivity that are the average of the permeabilities and permittivities ofthe layers. In [57], experimental results supporting this interpretation were provided.

Results reported in this section show the possibilities of ferrites magnetized to sat-uration as constituent elements of left-handed metamaterials. Magnetized ferriteshave the drawbacks of being highly anisotropic, fragile, and expensive. They alsoneed an external magnet, which may prevent many technological applications.However, they have the key advantage of tunability by a proper adjustment of theexternal biasing field. Because most bulk metamaterials are resonant devices, witha narrow frequency band of operation, such tunability can be a crucial advantage.

2.5.2 Chiral Metamaterials

Optical activity in natural materials is known in physics from the former analysis ofArago and Biot, in the early 19th century. After Pasteur’s studies on the crystal struc-ture of optically active materials, it became apparent that optical activity is closelyrelated to handedness or chirality. As has already been mentioned in Section 2.3.2,if the constituents of a material are not invariant by inversion (that is, if they havesome handedness), cross-polarization effects can be expected in the material.Therefore, isotropic optically active media can be considered as a particular case ofthe wider category of bianisotropic media. In fact, the constitutive relations for an iso-tropic and optically active medium are quite similar to equations (2.56) and (2.57):

D ¼ 1Eþ jffiffiffiffiffiffiffiffiffiffi10m0

pkH; 1 ¼ 10(1þ xe) (2:114)

B ¼ �jffiffiffiffiffiffiffiffiffiffi10m0

pkEþ mH; m ¼ m0(1þ xm), (2:115)

where xe, xm, and k are the electric, magnetic, and magnetoelectric susceptibilities,respectively. For lossless media, these quantities are real numbers [22].

The term chiral media is usually restricted to artificial materials that replicatenatural optical activity at microwave frequencies [22]. The first systematic experi-ments on chiral media were reported by Lindmann between 1914 and 1920 [67].Lindmann placed several hundreds of small helices made of copper wire insidecotton balls, and then put these balls, randomly orientated, inside a cardboard box.When these boxes were illuminated by a linearly polarized electromagnetic wave(of a frequency between 1 and 3GHz), the rotation of the plane of polarizationafter crossing the box was clearly observed. This rotation changed sign when thehandedness of the helices was reversed, and disappeared when a racemic (balanced)mixture of helices of both handedness was put into the box. Other inclusions also


producing chirality were later proposed in [68] to [70].26 Because all these inclusionsshow a resonant behavior, strong values of the susceptibilities xe, xm, and k can beexpected around the resonance. Thus, causality constraints suggest that regions ofsimultaneously strong and negative dielectric and magnetic susceptibilities shouldappear just above the resonance. In fact, negative refraction from artificial chiralinclusions has been proposed either for chiral mixtures [71,72] or for racemicmixtures of such elements [73].

Plane waves in biisotropic chiral media are TEM circularly polarized waves, withsquare wavevectors [22]

k2 ¼ k20(ffiffiffiffiffiffiffiffiffimr1r

p+ k)2; 1r ¼ 1=10, mr ¼ m=m0, (2:116)

where k0 ¼ vffiffiffiffiffiffiffiffiffiffim010

pis the wavevector in free space, and the sign of the square roots

is chosen positive by convention (in fact, a change in this sign will not affect theequation). The four roots of equation (2.116) give the four RCP and LCP wavespropagating in the positive and negative directions, respectively. The TEM waveimpedance for all these waves is the same [22]:

h ¼ffiffiffiffim

1

r: (2:117)

In general, chiral media present forbidden frequency bands for plane-wave propa-gation at those frequencies where 1 and m have opposite sign. At such frequencies,the propagation constant (2.116) becomes complex, and the plane wave becomes anonpropagating “complex wave”.27 The same condition holds for racemic mixturesof chiral particles. However, in this case, the wavevector k ¼ v

ffiffiffiffiffiffi1m

pbecomes imagin-

ary and the plane wave is evanescent in the forbidden frequency region. Only mediawith 1m . 0 at all frequencies do not present forbidden frequency bands. This con-dition is fulfilled when xe ¼ xm for all frequencies. In such a case, the wave impe-dance (2.117) coincides with that of free space.28

The conditions for backward-wave propagation—and therefore for negative refrac-tion—in chiral media have been analyzed in [72]. They can be deduced from the con-dition

k � S , 0, (2:118)

26As has already been mentioned, any resonant inclusion without inversion symmetry must show somedegree of cross-polarizability, leading to a bianisotropic medium. Chiral inclusions can be considered asa particular case of the former inclusions, showing resonant electric, magnetic, and cross-polarizabilitiesalong the same direction.27Complex waves are plane waves that presents a complex propagation constant even if the media is loss-less. They can appear in bi(iso/aniso)tropic media, as well as in some complex guiding systems [61].28However, except for normal incidence, this does not mean that there will not be reflected waves at theinterface between free space and such a chiral medium. The reason for this behavior is the differentvalues taken by the wavevectors in free space and in the chiral medium.


where k is the wavevector and S is the real part of the Poynting vector for the planewave. From this equation, cumbersome but straightforward calculations lead to thefollowing general condition [72]:

ffiffiffiffiffiffiffiffiffi1rmr

p+ k , 0, (2:119)

where the negative sign of the square root has to be chosen if 1r and mr are both nega-tive. According to equation (2.119), if k2 , j1rmrj then both solutions of equation(2.116) show negative refraction when both 1r and mr are negative. However, ifk2 . j1rmrj, only one of the solutions of equation (2.116) can be a backward waveand therefore will experience negative refraction at the interface with ordinary media.

Concerning racemic mixtures, it is well known that they present backward-wavepropagation when 1 , 0 and m , 0 simultaneously.

In order to obtain a better understanding of the behavior of artificial chiral media,we will analyze the quasiplanar inclusion proposed in [74] and shown in Figure 2.31.It is a slight modification of the two-turns spiral resonator shown in Figure 2.11. Thismodification is enough to provide a chiral behavior, while maintaining the equivalentcircuit shown in Figure 2.11. In particular, the frequency of resonance is still given by

v0 ¼ 1=ffiffiffiffiffiffiLC

p; C ¼ 2prCpul, (2:120)

where L is the ring inductance (2.152), Cpul is the per unit length capacitance betweenthe rings (2.161), and r the mean radius of the inclusion. However, unlike the planarspiral of Figure 2.11, the quasiplanar helix of Figure 2.31 presents quasistatic cross-polarizations. Such cross-polarizations can be obtained from an analysis similar tothat reported in Section 2.3.1. For lossless inclusions, the result is

mz ¼ ammzz Bext

z � aemzz E

extz (2:121)

and

pz ¼ aeezzE

extz þ aem

zz Bextz , (2:122)

FIGURE 2.31 Quasiplanar chiral resonator.


where the polarizabilities are given by

ammzz ¼ p2r4

L

v20

v2� 1

� ��1

, (2:123)

aemzz ¼ +jpr2tC0

v20

v

v20

v� 1

� ��1

, (2:124)

and

aeezz ¼ t2C2

0Lv40

v2

v20

v2� 1

� ��1

, (2:125)

where C0 is the in vacuo capacitance between the rings (C0 � 10=1C), and the sign inequation (2.124) depends on the helicity of the element. These expressions are, infact, very similar to equation (2.35) to (2.38), and the presence of the in vacuo capaci-tance C0 is justified as in the footnote of Section 2.3.1. An interesting property of theabove polarizabilties is that

ammzz aee

zz ¼ �(aemzz )

2: (2:126)

This property is very similar to equation (2.40). It is quite general, being shared bymany chiral inclusions, and comes from its LC resonant nature (see Problem 2.15).In addition to these resonant polarizabilities, the particle also shows some non-resonant quasistatic polarizabilities aee

xx and aeeyy, which can be estimated as (see

Section 2.3.1)

aeexx ¼ aee

yy ¼ 10163r3ext, (2:127)

where rext is the external radius of the particle. Therefore, for a random arrangementof particles, the average isotropic polarizabilities per particle are

kamml ¼ 13ammzz , (2:128)

kaeml ¼ +13aemzz , (2:129)

and

kaeel ¼ 13aeezz þ

23aeexx (2:130)


In order to develop an ab initio analysis of a random arrangement of the con-sidered inclusions, we will assume a zero-order homogenization:29

xe ¼110

Nkaeel; xm ¼ m0 Nkamml; k ¼ +j


10

rNkaeml, (2:131)

where N is the number of inclusions per unit volume. If the medium is made from abalanced (or racemic) mixture of elements, it will be isotropic with k ¼ 0.

The electric susceptibility xe can be expanded into two terms, one resonant andother nonresonant:

xe ¼ xnre þ xre ; xnre ¼ 32N910

r3ext ; xre ¼1310

Naeezz : (2:132)

From equation (2.131) it follows that a property quite similar to equation (2.126)also holds for the resonant susceptibilities:

xrexm ¼ k2: (2:133)

Near the resonance, where xer takes very high values, the nonresonant susceptibility is

much smaller than the resonant one.30 Therefore, the above inequality becomes

xexm � k2: (2:134)

As has already been shown, media with xe � xm minimize the forbidden fre-quency bands. From equation (2.134), such a condition also implies that

xm � xe � jkj: (2:135)

The corresponding condition for the resonant polarizabilities (2.123) to (2.125) is

c2aeezz ¼ amm

zz ¼ +jcaemzz ; c ¼ 1ffiffiffiffiffiffiffiffiffiffi

10m0p , (2:136)

29We know from Section 2.3.3 that this simple homogenization can give a good qualitative picture of thebehavior of the system. A careful calculation (not included) shows that a similar behavior is obtained ifLorentz local field theory is applied.30 This condition is not too restrictive in practice. For instance, if the ratio between the external diameter ofthe particle and the average distance between particles is 2/3, the resulting nonresonant susceptibility isxnre ¼ 0:13.


which is compatible with equation (2.126). Taking into account equation (2.134),condition (2.119) can be rewritten as

1þ xe þ xm , 0: (2:137)

Let us consider the particular case when equation (2.135) and (2.136) are fulfilled. Insuch a case the forbidden frequency band is minimized, and backward-wave propa-gation occurs for

xe � xm .�0:5: (2:138)

In this region, k . 1rmr and, therefore, only one of the solutions of equation (2.116)exhibits negative refraction.

At frequencies where xe � xm � �0:5, k � 0 for the corresponding wave.Therefore, at such frequencies there is power transmission (or nonzero group velocity)with very small (or even zero) phase velocity. This behavior recalls that of thebalanced right/left-handed structures (analyzed in Chapter 3).

For racemic dilute mixtures of the same chiral particles, backward-wave propa-gation is present in the frequency range

xe � xm . �1, (2:139)

and the aforementioned balanced behavior occurs for xe � xm � �1. Therefore, thefrequency band for backward-wave propagation is wider in biisotropic mixturesthan in racemic ones. The price to pay for this bandwidth enhancement is that onlyone of the two circular polarizations shows this behavior in the reported biisotropicmixtures.

Negative refractive metamaterial design from random (chiral or racemic) mixturesof chiral inclusions is advantageous because left-handedness comes from only onekind of element. Regarding the quasiplanar chiral inclusion analyzed in thissection, its main advantage is the reliability and reproducibility that can be obtainedfrom standard photoetching manufacturing techniques. Other works dealing with theapplication of chirality to left-handed media design have stressed the usefulness of amixture of a chiral medium operating far from the resonance and a resonant dielectricmedium [75]. Finally, the specific situation when 1=10 ¼ m=m0 ¼ 0 but k = 0, so-called chiral nihility, has been analyzed in [76].

2.5.3 Other Proposals

In this section we will briefly describe some alternative designs to bulk left-handedmetamaterials. Because a description of all such alternatives would be imposible(and the number of such proposals steadily increase with time), this descriptionwill be necessarily incomplete.


An interesting alternative for the design of left-handed metamaterials at microwavefrequencies has recently been proposed in [78] and [79]. This proposal starts from theconsideration of a system of spheres embedded in a dielectric matrix. The problem ofthe homogenization of an array of dielectric or magnetic spheres inside a homo-geneous host matrix has long been known. If the size of the spheres is small comparedwith the wavelength in both the host medium and the spheres, the array behaves as ahomogeneous medium with intermediate values of 1 and m [77]. However, if the sizeof the spheres is of the same order as the internal wavelength, they may present res-onances at some specific frequencies (such resonances are sometimes referred to asMie resonances). If the wavelength in the host medium is still larger than the sizeof the spheres, then the array can be seen as a homogeneous medium, with a resonantpermittivity or permeability that, eventually, may be negative above the resonances[78,79]. In order to achieve this behavior, at least one of the constitutive parametersof the spheres, 1s and/or ms, must be much higher than the corresponding constitutiveparameter of the host medium 1h and/or mh:

1s

1h� 1 and/or

ms

mh

� 1: (2:140)

The problem of the homogenization of a system of spheres embedded in a hostmedium satisfying (2.140) was studied by Lewin [80] in 1947. Lewin providedclosed expressions for the constitutive parameters of the homogenized medium,which were subsequently utilized in [78] and [79] for their analysis.

Starting from Lewin’s results, Holloway et al. [78] analyzed a hypothetical com-posite medium made of magnetodielectric spherical particles embedded in a hostmedium, with 1h � 1s and mh � ms. They found that, for 1s and ms high enough,the homogenized medium shows a frequency band of simultaneously negative 1

and m. However, isotropic media showing simultaneously very high electric permit-tivity and magnetic permeability are not known for microwaves. In fact, even isotro-pic media with very high magnetic permeability are not known for microwaves.However, ferroelectric or ceramic media with very high dielectric constants are avail-able in such a frequency range. Vendik and Gashinova [79] considered this specificcase, that is, 1s � 1h, ms ¼ mh ¼ m0. Dielectric resonators made with smallinclusions of high dielectric constant, such as the spheres proposed in [79], arewell known in microwave technology. They are usually analyzed by assuming “mag-netic wall” boundary conditions around the resonator,31 that is, assuming the follow-ing conditions at the sphere’s boundary:

n � Es ’ 0; nHs ’ 0; (2:141)

where n is the normal to the interface, and Es,Hs are the fields inside the sphere. Fromthis approach it follows that the field distribution inside the spheres corresponds

31This assumption can be justified from the fact that the reflection coefficient at the boundary between amedium of high permittivity and a medium of low permittivity is approximately þ1, the same as for amagnetic wall [81].


approximately to that of a spherical cavity in a perfect magnetic conductor. Thisproblem is dual to that of a spherical cavity inside a perfect electric conductor, awell-known problem of classical electromagnetism [82]. Figure 2.32 shows the elec-tric and magnetic field inside the high-permittivity spheres proposed in [79], obtainedfrom the aforementioned magnetic wall approximation. It becomes apparent fromsuch a figure that the first resonance (TE011 resonance) is associated with a magneticdipole, arising from the circulation of the electric displacement current. The secondresonance (TM011) is associated with an electric dipole arising from the continuityof the tangential electric field at the sphere’s boundary. Around such resonances,the aforementioned dipoles must show the universal causal behavior, whichensures that they are positive/negative (regarding the external excitation) below/above the resonance. Starting from this result, Vendik and Gashinova [79] proposeda composite left-handed medium made of two sets of high-permittivity dielectricspheres of different radii, embedded in a dielectric matrix. The radius of each set ischosen so that the first and the second Mie resonances are excited at each set ofspheres at the same frequency. Therefore, according to causality requirements, justabove this specific frequency the permittivity and the permeability of the effectivemedium must be negative. Numerical finite difference time domain (FDTD) simu-lations supporting this conclusion were reported in [79].

The aforementioned proposal is conceptually simple, and has the additionaladvantage of being intrinsically isotropic. The main disadvantage of such a designseems to be the small bandwidth, about a 1% in the analyzed structures [79]. A limit-ation of this proposal comes from the fact that, the higher the 1s, the smaller the inter-action between the fields internal and external to the spheres (in the limit 1s ! 1, thespheres behave as perfect conducting spheres).

Negative-permittivity mediamade from systems ofmetallic wires or plates present animportant drawback for small absolute values of 1, such as those needed for the design ofsuper-lenses and other devices of practical interest. In fact, according to equations (2.2)

FIGURE 2.32 Sketch of the magnetic (solid lines) and electric (dashed lines) fields inside aspherical cavity in a perfect magnetic conductor, for the two first resonances. (a) TE011 firstresonance. (b) TM011 second resonance. (Source: Reprinted with permission from [79]; copy-right 2003, IEEE.)


and (2.8), the wire spacing must be of the same order as a half-wavelength in order toobtain values of 1=10 � �1. Therefore, the continuous-medium approach can behardly justified for such values of 1 (see Section 2.4.3). In order to overcome thisdifficulty, a modification of the SRR concept has been proposed in [83]. The proposedstructure, two symmetric closed inductive loops connected to a common capacitor, isshown in Figure 2.33a. The equivalent circuit is shown in Figure 2.33b. At resonance,the magnetic moment associated with currents in both loops cancels, only the electricdipole associated with the capacitor remaining. According to causality requirements,the electric polarizability must be large and negative just above the resonance, thusproviding the required behavior for negative-permittivity metamaterial design. Byproperly adjusting the resonator parameters (by slight modifications of the basicdesign [83]) the electrical size of the resonator can be made small enough to overcomethe aforementioned drawback of wire media. In [83], numerical calculations supportingthe above discussion are reported. In our opinion, the main inconvenience of thisproposal may be the excitation of unwanted magnetoinductive waves (see Chapter 5)due to the magnetic coupling between adjacent resonators.

It was shown in Section 2.2.1 that a hollow metallic waveguide operating in thefundamental TE10 mode behaves as a one-dimensional ideal plasma, with plasma fre-quency equal to the cutoff frequency of the waveguide. Let us now consider a rec-tangular waveguide operating at the first TM mode, that is, the TM11 mode. Thepropagation constant is still given by equation (2.1); however, the wave impedanceis now given by

Z ¼ k

v10, (2:142)

instead of equation (2.3). Therefore, from the same rationale as in Section 2.2.1, itfollows that the waveguide behaves now as a magnetic plasma with effective dielec-tric constant

meff ¼ m0 1� v2c

v2

� �, (2:143)

FIGURE 2.33 (a) Electric coupled-field resonator proposed in [83] for negative permittivitymetamaterial design: (a) Basic configuration; (b) Equivalent circuit.


and effective dielectric constant 1eff ¼ 10. This concept was proposed and demon-strated by Esteban et al. [11] by filling a below-cutoff square waveguide with atwo-dimensional wire medium of effective negative dielectric constant given byequation (2.2).32 The proposed structure is shown in Figure 2.34. The propagationconstant for TM modes is given by [11]

k2 ¼ v210m0 1� v2c

v2

� �1�

v2p

v2

!, (2:144)

where vc is the cutoff frequency of the TMmode of the hollow waveguide, and vp theplasma frequency of the wire system. It can be easily realized that propagation isbackward for v , vc, vp (which correspond to the left-handed passband) andforward for v , vc, vp. For vc , v , vp (or vc . v . vp) there is a stopbandof negative 1 (or m). A key advantage of this proposal over other one-dimensionalleft-handed media waveguide simulations (see Section 2.4.1) is a wider bandwidth,as well as smaller losses due to the absence of resonances in the system. The casewith vc ¼ vp is of interest because no stopband appears. The reported behavior isquite similar to the behavior of the composite left-/right-handed transmission lines[5–7], which will be analyzed in detail in the next chapter. In fact, the proposal ofEsteban et al. [11] can be seen as the waveguide counterpart of the aforementionedproposals.

To end this section we will devote some words to the synthesis of left-handedmetamaterials at optical frequencies. Because negative permittivity is alreadypresent at such frequencies in many metals, the main challenge seems to be how toobtain negative permeability. That materials do not show a magnetic response

FIGURE 2.34 Simulation of a one-dimensional left-handed medium by a below-cutoff TMhollow waveguide, filled by a system of crossed wires [11].

32In fact, the analysis in [11] models the system as a hollow waveguide filled with a uniaxial dielectricmedium with 1? , 0 and 1k ¼ 10 (see Problem 1.16 of Chapter 1).


at optical frequencies has been almost a dogma in optics for many years. The reasonusually invoked to support this statement is that optical frequencies are of the sameorder as natural frequencies of oscillation of electrons in atoms and molecules.Therefore, it seems impossible to obtain closed loops of atomic currents—andtherefore magnetization—at such frequencies [13]. More recently, many attemptshave been made in order to obtain a nonvanishing magnetic response at infraredand optical frequencies from artificial atoms, such as SRRs [39] and other nanofab-ricated plasmonic structures [84,85], which can also be modeled as LC resonantcircuits [86]. However, the measured magnetic response of such structures wasalways much smaller than those obtained from similar structures at microwave fre-quencies. This result is in agreement with the considerations already made inSection 2.3.6.

APPENDIX

In this appendix, accurate quasianalytical expressions for the inductances and per unitlength capacitances used in Section 2.3 will be provided. To begin, the self-induc-tance of a ring of mean radius r and width c will be obtained. Let us assume auniform current I on the ring. As there is no field and/or current dependence onthe azimuthal coordinate, the magnetic energy can be computed as

UM ¼ p

ð10

rAfJs,f dr, (2:145)

where Af is the azimuthal component of the vector magnetic potential. Integrating byparts and using Bz ¼ r�1@r(rAf), where Bz is the z component of the magnetostaticfield, and @r the partial derivative with respect to r, it is found that

UM ¼ p

ð10

rBzI(r) dr, (2:146)

where I(r) is defined by

I(r) ¼ð1r

Js,f(r0) dr0, (2:147)

and Js,f is the azimuthal surface current density on the ring.Because the currents are restricted to the z ¼ 0 plane, Bz can be derived from

a scalar magnetic potential, c(r, z): Bz ¼ �m0@zc. This scalar magnetic potentialmust satisfy Laplace’s equation, @2

zcþ r�1@r(r@rc) ¼ 0, subjected to the follow-ing boundary conditions: @zc(r, 0þ) ¼ @zc(r, 0�); I(r) ¼ c(r, 0þ)� c(r, 0�) and

APPENDIX 107

c(1, z) ; c(r,1) ¼ 0. Taking the Fourier–Bessel transform, which is defined as

eF(k) ¼ð10

rJ0(kr)F(r) dr, (2:148)

the above problem is analytically solved for ~c(k, z), which is found to be

ec (k, z) ¼ +12eI(k)e+kz, (2:149)

where the upper (lower) sign stands for z . 0 (z , 0).After introducing Bz ¼ �m0@zc into equation (2.146), using equations (2.148)

and (2.149), and making use of the Parseval theorem as well as the relationL ¼ 2UM=I2, the SRR inductance is obtained as

L ¼ m0p2

I2

ð10

[~I(k)]2 k2 dk: (2:150)

For practical computations, a constant value for Js,f has been assumed on the ring,that is,

Js,f ¼ I=c for r � c=2 , r0 , r þ c=20 otherwise.

�(2:151)

Introducing this expression into equation (2.147), after some algebraic manipulations,the SRR inductance can be obtained as the following integral:

L

m0¼ p3

4c2

ð10

1k2

(bB(kb)� aB(ka))2 dk, (2:152)

where a ¼ r � c=2, b ¼ r þ c=2 and function B(x) is defined as

B(x) ¼ S0(x)J1(x)� S1(x)J0(x), (2:153)

with Sn and Jn being the nth order Struve and Bessel functions respectively. This inte-gral can be easily calculated in a computer, giving an accurate evaluation of the SRRinductance [19].

Regarding the per unit length capacitances for the EC-SRR and the BC-SRR, theycan be directly obtained from well-known microwave circuit design formulas. For theEC-SRR, from Table 2.7 and equation (2.4) of [20], it is directly obtained that

Cpul=10 ¼ 1eF(k), (2:154)


where

1e ¼ 1þ 1� 10

210

F(k)F(k1)

(2:155)

k ¼ a=b; a ¼ d=2; b ¼ d=2 þ c (2:156)

k1 ¼sinh(pa=2t)sinh(pb=2t)

(2:157)

F(k) ¼ 1pln 2

1þffiffiffiffik0

p

1�ffiffiffiffik0

p� �

(2:158)

if 0 � k � 0.7, and

F(k) ¼ p ln 21þ

ffiffiffik

p

1�ffiffiffik

p� �� 1

(2:159)

if 0.7 �k �1, where

k0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

p: (2:160)

For the BC-SRR, from Table 2.6 of [20], it directly follows that

Cpul=10 ¼121e

2ctþ 1:393þ 0:667 ln

2ctþ 1:444

� �� (2:161)

where

1e ¼1þ 10

210þ 1� 10

2101þ 6 t

c

� ��1

: (2:162)

PROBLEMS

2.1. Polarizability of a waveguide. Define the average polarization of a rectan-gular waveguide that supports the fundamental TE10 mode as

P ¼ 1jvSw

þJs dl, (2:163)

where Js is the surface current on the waveguide walls, Sw the surface of thewaveguide cross-section, and the integral is over the boundary of the

PROBLEMS 109

waveguide section. Show that this average polarization is related to the averageelectric field in the waveguide cross-section by equation (2.5).

2.2. TM waveguides as artificial magnetic plasmas. Show, following the sameprocedure as in Section 2.2.1, that a waveguide operating in a TM mode canbe seen as an artificial magnetic plasma with an effective dielectric constant1eff ¼ 10 and an effective permeability meff ¼ m0(1� v2

c=v2), where vc is

the cutoff frequency of the TM mode.

2.3. Spatial dispersion in wire media. Show that a uniaxial medium with permit-tivity tensor 1x ¼ 1y ¼ 10 and 1z(v) ¼ 10(1� v2

p=v2) gives the following

dispersion relation for TM (to z) waves:

k2x þ k2y ¼1z(v)10

(k20 � k2z ): (2:164)

Show that if 1z is given by equation (2.14), then the dispersion relation (2.13) isobtained.

2.4. Three-turns spirals as metamaterial elements. Consider the resonator shownin Figure 2.35. Derive an equivalent circuit for the determination of the fre-quency of resonance of this element. Using this equivalent circuit show thatthe frequency of resonance, f, of this element is related to the frequency ofresonance f2SR of the two-turns spiral (Fig. 2.11) of the same mean radius,strip width, and slot width through f ’ f2SR=

ffiffiffi2

p. Derive an expression for

the magnetic polarizability of such an element (see [26]).

2.5. Nonbianisotropic two-turns spirals. Consider the metamaterial elementshown in Figure 2.36. It can be easily realized that it presents inversion sym-metry, so that its cross-polarizabilities must vanish. Derive an equivalent circuit

FIGURE 2.35 Sketch of a three-turns spiral resonator. Metallizations are in gray, and thearrows show the direction of the ohmic and displacement currents at resonance.


for this element. Using this equivalent circuit, show that the frequency of res-onance, f, of this element is related to the frequency of resonance, f2SR, of thetwo-turns spiral (Fig. 2.11) of the same mean radius, strip width, and slot widththrough f ¼

ffiffiffi2

pf2SR. Derive an expression for the magnetic polarizability of

such element.

2.6. Open EC-SRR. Consider the configuration shown in Figure 2.37. Derive anequivalent circuit for this configuration. Using this equivalent circuit showthat this element is equivalent to a short circuit at a frequency that is a halfof the frequency of resonance of the EC-SRR of similar design. This configur-ation is useful for microwave filter design (see [87]).

2.7. Dispersion relation for plane TEM waves in SRR media. Consider the SRRmedia shown in Figure 2.12, with EC-SRRs as elementary constituents. Usingthe model reported in Section 2.3.3, show that the dispersion relation for elec-tromagnetic plane waves propagating along the x-axis and polarized with the

FIGURE 2.36 Sketch of a nonbianisotropic spiral resonator. Note the presence of two non-connected metallizations.

FIGURE 2.37 Open split rings resonator.

PROBLEMS 111

magnetic field parallel to the z-axis is

kx ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimzz1yy � m010k

2yz

q: (2:165)

2.8. Effects of bianisotropy on the dispersion relation of a left-handed medium.Using expression (2.76), and assuming the superposition hypothesis for theleft-handed composite of wires and SRRs reported in [4], find the width ofthe mismatch between the negative-m stopband and the left-handed passbandof Figure 2.22.

2.9. TM plane waves in SRR media. Consider the SRR media shown inFigure 2.12, with EC-SRRs as elementary constituents. Show that planewaves propagating along the y-axis and polarized with the magnetic fieldalong the z-axis are TM waves, with the electric field elliptically polarized inthe x–y plane. Show that the dispersion relation for such waves is

k ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffi1yymeff

p; meff ¼ mxx � 10m0

k2yx

1yy: (2:166)

2.10. Magnetostatic waves in SRR media. Consider the SRR medium shown inFigure 2.12, with BC-SRRs as elementary constituents. Consider a quasimag-netostatic field, H ¼ �rcm, where cm is the magnetostatic potential.

† Show that magnetostatic waves with the dispersion relation

k2x þ k2y þmzz

m0k2z ¼ 0 (2:167)

can propagate through the metamaterial in the region of mzz , 0:† Study the propagation of magnetostatic modes in a slab of the SRR mediumof width d. Consider the two cases, with the SRRs in the plane of the slab,and with the SRRs orthogonal to the plane of the slab.

These waves can be considered analogous to the well-known magnetostaticwaves in ferrites [58,59] (see also Section 2.5.1). As happens in ferrites, suchmagnetostatic approximation is only valid in the limit k � k0 ; v0


p.

2.11. Magnetic susceptibility of a nano-SRR media at near-optical frequencies.Consider a cubic network of SRRs such as that shown in Figure 2.12, withelementary constituents such as those shown in Figure 2.18b. Show that, iflosses are small, the magnetic susceptibility along the ring axis is approxi-mately given by

xm � v2m

v20 � v2 þ jvg

, (2:168)


where

v2m ¼ m0

p2rv20

8K3(Lm þ Lk), (2:169)

v0 ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(Lm þ Lk)Cp , g ¼ R

Lm þ Lc� v, v0, (2:170)

and K is the dimensionless ratio a/2r.Show that, when Lk . Lm, the maximum value of j<(xm)j is

jRe(xm)jmax �p3

8K3

v0

fc

S

l2p

!2

, (2:171)

where S is the wire section.

2.12. Magnetostatic surface modes in a saturated ferrite I. Show that themagnetostatic wave equation (2.111) has bounded propagative solutions atthe interface between a magnetized ferrite and a perfect conductor forv ¼ vH þ vM. These surface magnetostatic waves have the following charac-teristics: static magnetization H0 parallel to the surface and propagation alongthe direction defined by H0 n, where n is the outgoing normal to the ferriteinterface. As in quasistatic surface plasmons (see Chapter 1), the dispersionequation for such waves is degenerate (any wavevector is excited at theaforementioned frequency).

2.13. Magnetostatic surface modes in a saturated ferrite II. Show that the mag-netostatic wave equation (2.111) has bounded propagative solutions at theinterface between a magnetized ferrite and free space similar to those analyzed

FIGURE 2.38 A chiral inclusion made of two metallic balls, connected by a solenoid.

PROBLEMS 113

in the previous problem. Show that the frequency of excitation of such MSSWsis v ¼ vH þ 1

2vM.

2.14. Magnetostatic surface modes in a saturated ferrite III. Find, from Maxwellequations, the dispersion equation without approximations for the MSSWsanalyzed in the previous problem (see [58] for more information).

2.15. Quasistatic analysis of a chiral inclusion. Consider the chiral inclusion ofFigure 2.38. Using elementary formulas for the self-inductance of a solenoid,and for the capacitance between two metallic balls [9], find closed expressionsfor the polarizabilities of the inclusion.

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42. J. R. Reitz, F. J. Milford, and R. W. Christy Foundations of Electromagnetic Theory.Addison-Wesley, Reading, MA, (4th ed.).

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45. R. A. Shelby, D. R. Smith, and S. Schultz “Experimental verification of a negative index ofrefraction.” Science, vol. 292, pp. 77–79, 2001.

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CHAPTER THREE

Synthesis of Metamaterials inPlanar Technology

3.1 INTRODUCTION

This chapter is focused on the synthesis of left-handed and single negative structuresin planar technology. Essentially, one-dimensional propagating structures, that is,metamaterial transmission lines, will be considered, although a subsection will alsobe dedicated to the implementation of planar two-dimensional left-handed media.Regarding the synthesis of metamaterial transmission lines, most of the chapterwill be devoted to the resonant-type approach, where the formerly constitutive par-ticles for the implementation of bulk left-handed structures, that is, the SRRs, areused in combination with other elements. A resonant particle that is the dualversion of the SRR, which is useful for the implementation of negative-permittivitymedia in planar technology will also be introduced in this chapter, that is, the comp-lementary split rings resonator (CSRR). The lumped-element equivalent circuitmodels for left-handed and single negative transmission lines loaded with bothSRRs and CSRRs will be presented, analysed, and discussed. These models are fun-damental for the design of microwave components based on SRRs and CSRRs, aswill be shown in Chapter 4.

This book predominantly reflects the results of the research activities carried outby the authors and their respective groups in the field of metamaterials. These activi-ties have been mainly oriented towards the resonant-type approach, and the authorshave pioneered the synthesis and applications of metamaterial transmission linesbased on SRRs, CSRRs, and related topologies. However, chronologically, therewas an earlier approach to the design of metamaterials in planar technology,namely that based on the dual transmission line concept. This approach, initially pro-posed by Iyer et al. [1], Oliner [2] and Caloz et al. [3], has reached a level of maturity


119

that has led to many prospective engineering applications in the field of microwaves.Therefore, the first sections of this chapter will be devoted to a brief introduction ofthis approach. Applications of both approaches (dual transmission line and resonant-type approaches) will be discussed in Chapter 4.

3.2 THE DUAL (BACKWARD) TRANSMISSION LINE CONCEPT

The concept of backward waves to describe propagating waves with antiparallelphase and group velocities is not new (several textbooks have dealt with anddescribed this subject [4,5]). Conceptually, backward waves can be generated byfeeding a ladder network with alternating shunt-connected inductors and seriescapacitors (Fig. 3.1). Such a network is the dual version of the equivalent-circuitmodel of a conventional lossless planar transmission line with forward (as opposedto backward) wave propagation (see also Fig. 3.1). The analysis of the propagationcharacteristics of the structures shown in Figure 3.1 can be realized from thetheory of periodic structures [6], where it is assumed that the structure is either infiniteor it is matched to the ports. From this analysis, the phase constant,1 b, and thecharacteristic (or Bloch) impedance,2 ZB, of the transmission media can be inferredaccording to

cosbl ¼ 1þ Zs(v)Zp(v)

(3:1)

ZB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZs(v)[Zs(v)þ 2Zp(v)

p], (3:2)

where Zs and Zp are the series and shunt impedances of the basic cell of the structure,described by its T-circuit model, and l is the period. Application of expressions (3.1)and (3.2) to the circuits of Figure 3.1 leads to the following expressions [the sub-indices L and R are used to distinguish between the left-handed (backward wave)

1The phase constant is usually designated by b in circuit theory, whereas in the previous chapters it hasbeen designated by k, which is the usual convention to express the dispersion relation in continuous media.2The characteristic impedance of a transmission line is the relation between voltage and current for a singlepropagating wave at any position of the line. In periodic structures (either infinite or matched to the ports)the relation between voltage and current for the Bloch waves is usually called Bloch impedance. For finiteperiodic structures not matched to the ports, both incident (energy flow from the input to the output port)and reflected (energy flow in the opposite direction) waves necessarily arise. For either propagating wave(incident or reflected), expression (3.2) provides the relation between voltage and current. This impedanceis usually called image impedance, as it is the load impedance required to see identical input impedance.Nevertheless, we will not distinguish between image and Bloch impedances, and we will call the impe-dance given by expression (3.2) Bloch impedance or, even, characteristic impedance. Indeed the imageor Bloch impedance play exactly the same role as the characteristic impedance in conventional lines.

120 SYNTHESIS OF METAMATERIALS IN PLANAR TECHNOLOGY

and right-handed (forward wave) structures]:

cosbRl ¼ 1� LC

2v2 (3:3)

ZBR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

C1� v2

v2cR

� �s(3:4)

cosbLl ¼ 1� 12LCv2

(3:5)

ZBL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

C1� v2

cL

v2

� �s(3:6)

where vcR ¼ 2=(LC)1=2 and vcL ¼ 1=2(LC)1=2 are angular cutoff frequencies for theforward and backward wave transmission line structures, respectively. The dispersiondiagram as well as the dependence of ZB on frequency are depicted in Figure 3.2.Transmission is limited to those frequency intervals that make the phase constantand the characteristic impedance to be real numbers. It is worth mentioning that fre-quency dispersion is present in both structures. Even though the circuit of Figure 3.1bmodels an ideal lossless forward transmission line, where dispersion is absent, in factthis circuit is only valid for frequencies satisfying v � vcR, that is, in the long wave-length limit (corresponding to those frequencies where the wavelength for guidedwaves satisfies, lg � l ).3 To correctly model an ideal lossless transmission line athigher frequencies, we simply need to reduce the period of the structure, and, accord-ingly, the per-section inductance and capacitance of the line, L and C, with the resultof a higher cutoff frequency. Thus, the circuit of Figure 3.1b can properly describe

FIGURE 3.1 Equivalent circuit model of backward (a) and forward (b) transmission lines.The T-circuit models of the basic cell structures are also indicated in (c) and (d ).

3lg has been called the internal wavelength and was designated li in the previous chapter (see Section2.4.3).

3.2 THE DUAL (BACKWARD) TRANSMISSION LINE CONCEPT 121

ideal transmission lines without dispersion. To this end we simply need to select theperiod such that the long wavelength limit approximation holds. Under this approxi-mation, expressions (3.3) and (3.4) lead us to the following well-known expressions:

bR ¼ vffiffiffiffiffiffiffiffiffiL0C0

p(3:7)

and

ZBR ¼ffiffiffiffiffiL0

C0

r; Zlw, (3:8)

where L0 and C0 are the per-unit length inductance and capacitance of the trans-mission line (in expression 3.8, L0 and C0 can be replaced by L and C with noeffect). From (3.7), we can obtain the phase and group velocities of the forward trans-mission line. These velocities are given by:

vpR ¼ v

bR

¼ 1ffiffiffiffiffiffiffiffiffiL0C0

p ¼ lffiffiffiffiffiffiLC

p (3:9)

vgR ¼ @bR

@v

� ��1

¼ vpR (3:10)

and they are both positive and constant.

FIGURE 3.2 Typical dispersion diagram of forward (a) and backward (b) transmission linemodels. The dependence of the normalized (ZB/Zlw) Bloch impedance with frequency isshown in (c) and (d ) for the forward and backward lines, respectively.


Conversely, the backward-wave structure of Figure 3.1a is dispersive even in thelong wavelength limit. In order to identify this structure as an effective propagatingmedium (i.e., one-dimensional metamaterial), operation under this approximationis required.4 However, regardless of the operating frequency within the transmissionband, the structure supports backward waves and, for this reason, it is a left-handedtransmission line. In order to properly identify effective constitutive parameters, meff

and 1eff, the long wavelength approximation is necessary. However, many circuitapplications of these left-handed lines are based on left-handedness, rather than onthe effective medium properties, and operation under this approximation is not adue. Nevertheless, for coherence and simplicity, the phase constant, the characteristicimpedance, as well as the phase and group velocities are derived under the long wave-length limit (v � vcL). The following results are obtained:

bLl ¼ � 1

vffiffiffiffiffiffiLC

p (3:11)

ZBL ¼ffiffiffiffiL

C

r; Zlw (3:12)

vpL ¼ v

bL

¼ �v2lffiffiffiffiffiffiLC

p, 0 (3:13)

vgL ¼ @bL

@v

� ��1

¼ þv2lffiffiffiffiffiffiLC

p. 0 (3:14)

and the phase and group velocities have opposite signs. From a mathematical point ofview, in the dispersion diagrams for the forward and backward lines depicted inFigure 3.2, and obtained from expressions (3.3) and (3.5), the sign of b can beeither positive or negative. This ambiguity comes from the two possible directionsof energy flow, namely from left to right or vice versa. If we adopt the usual conven-tion of energy flow from left to right, then the sign of the phase constant is determinedby choosing that portion of the curves that provide a positive group velocity (boldlines in Fig. 3.2a and b). From this, it is clear that for the forward transmissionline, b and vp are both positive, whereas these magnitudes are negative for the back-ward transmission line. In both cases, vg is positive, as one expects, on account of thecodirectionality between power flow and group velocity.

In the backward and forward transmission lines depicted in Figure 3.1, it is poss-ible to identify effective constitutive parameters. To this end, we should take intoaccount that transverse electromagnetic (TEM) mode propagation in planar trans-mission media and plane wave propagation in isotropic and homogeneous dielectricsare described by identical equations (telegraphist’s equation) provided the following

4This aspect was discussed in Section 2.4.3.


mapping holds:

Z 0s(v) ¼ jvmeff (3:15)

Y 0p(v) ¼ jv1eff (3:16)

where Z0s and Y 0p are the series impedance and shunt admittance per unit length. Thus,

for the forward line, the effective permittivity and permeability are constant and theyare given by

1eff ¼ C=l (3:17)

and

meff ¼ L=l, (3:18)

whereas for the backward transmission line, the constitutive parameters are

1eff ¼ � 1v2Ll

(3:19)

and

meff ¼ � 1v2Cl

, (3:20)

and they are both negative, a sufficient condition to obtain left-handed wavepropagation.

From a practical point of view, in order to implement a dual (or backward) trans-mission line, a host line (microstrip or CPW, among others) is required. The host linethus introduces parasitic elements that in general may not be negligible, and theyshould be taken into account to accurately describe the propagation characteristicsof the lines. As will be shown later, these structures may exhibit left-handed orright-handed wave propagation, depending on the frequency interval, and, for thesereason, they have been termed as composite right-/left-handed (CRLH) transmissionlines [7]. The equivalent circuit model of these structures is depicted in Figure 3.3.For clarity, we have renamed the reactive elements of the dual transmission line asCL and LL, and CR and LR correspond to the per-section capacitance and inductanceof the host line. By using expressions (3.1) and (3.2), the dispersion relation as well asthe characteristic impedance of the CRLH transmission line can be inferred, namely:

cosbl ¼ 1� v2

2v2R

1� v2s

v2

� �1�

v2p

v2

!(3:21)

ZB ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLRCR

1� v2s

v2

� �

1�v2p

v2

!� L2Rv2

41� v2

s

v2

� �2

vuuuuuut (3:22)


where the following variables

vR ¼ 1ffiffiffiffiffiffiffiffiffiffiffiLRCR

p (3:23)

vL ¼ 1ffiffiffiffiffiffiffiffiffiffiffiLLCL

p (3:24)

and the series and shunt resonance frequencies

vs ¼1ffiffiffiffiffiffiffiffiffiffiffiLRCL

p (3:25)

vp ¼1ffiffiffiffiffiffiffiffiffiffiffiLLCR

p (3:26)

have been introduced to simplify the mathematical formulas. Expressions (3.21 and3.22) are depicted in Figure 3.4. Two propagating regions, separated by a gap at thespectral origin, can be distinguished. In the lowest frequency region the parameters ofthe dual transmission line, CL and LL, are dominant, and wave propagation is back-ward. This situation is reversed above the stopband, where the parasitic reactances ofthe host line make the structure behave as a right-handed line. Indeed, at high frequen-cies, the CRLH transmission line tends to behave as a purely right-handed (PRH)line. Conversely, in the lower limit of the first allowed band, the CRLH structure exhi-bits the characteristics of a purely left-handed (PLH)—or dual—transmission line.The gap limits are given by the frequencies satisfying.

vG1 ¼ min(vs, vp) (3:27)

vG2 ¼ max(vs, vp): (3:28)

In the long wavelength limit, expression (3.21) is rewritten as

b ¼ s(v)l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

v2R

1� v2s

v2

� �1�

v2p

v2

� �s, (3:29)

FIGURE 3.3 Equivalent circuit model (basic cell) of the CRLH transmission line.


where s(v) is the following sign function:

s(v) ¼ �1 if v , min(vs, vp)þ 1 if v . max(vs, vp):

�(3:30)

From the phase constant (equation 3.29), the phase and group velocities can be easilyinferred, these velocities being of opposite sign in the left-handed band and bothbeing positive in the right-handed band.

With regard to the constitutive parameters for the CRLH transmission line, theycan be inferred as previously indicated, that is,

1eff ¼CR

l� 1v2LLl

(3:31)

and

meff ¼LRl� 1v2CLl

, (3:32)

and they can be positive or negative, depending on the frequency range.

FIGURE 3.4 Typical dispersion diagram (a) and variation of Bloch impedance with fre-quency (b) in a CRLH transmission line model. In this example, vs , vp.


One particular case of interest is the so-called balanced CRLH line, which corre-sponds to the situation where the series and shunt resonances are identical, namelyvs ¼ vp ¼ vo [8]. In this case, there is not a forbidden band between the left-handedand right-handed allowed bands; in other words, the change between backward andforward wave behavior is continuous. Concerning the characteristic impedance, itreaches its maximum at vo (the transition frequency), where

ZB ¼ffiffiffiffiffiffiLRCR

r¼

ffiffiffiffiffiffiLLCL

r, (3:33)

and the characteristic impedance decreases as frequency increases or decreases fromvo. At the limits of the allowed propagation interval (which contains both the left-handed and right-handed frequency bands) the characteristic impedance nulls, andbeyond this limits, it takes imaginary values. Figure 3.5 depicts the dispersiondiagram and the variation of line impedance with frequency. In contrast, the charac-teristic impedance dependence on frequency is more complicated for the unbalancedcase (see Fig. 3.4). If we consider that vs , vp (which is the situation considered inFig. 3.4), the characteristic impedance is zero at the lower limit of the left-handedinterval, it increases as frequency increases, reaches a maximum, and then decreases

FIGURE 3.5 Typical dispersion diagram (a) and variation of Bloch impedance with fre-quency (b) in a balanced CRLH transmission line.


towards zero, at vs, the upper limit of the left-handed band. In the right-handed band,the impedance exhibits a pole at vp (the lower frequency limit), it decreases as fre-quency increases, and it nulls at the higher cutoff frequency of the right-handedband. However, the behavior changes if vs . vp; in this case, the impedance goesfrom zero up to infinity in the left-handed passband, and it nulls at the edges ofthe right-handed band.

From the point of view of the characteristic impedance, the balanced CRLH trans-mission line is interesting because in the vicinity of the transition frequency, ZB is notvery dependent on frequency, and it allows for broadband matching, as compared tothe unbalanced case, where the characteristic impedance is much more sensitive tofrequency. On the other hand, the balanced case exhibits another important differenceas compared to the unbalanced CRLH transmission line. Namely, at the transition fre-quency, where the phase velocity exhibits a pole, the phase shift5 is zero, and thegroup velocity differs from zero. In other words, at this frequency, wave propagationis possible (vg = 0) with b ¼ 0. The phase origin of the balanced CRLH line thustakes place at vo. When frequency is decreased below vo, the phase becomes positiveand it increases progressively; conversely, when frequency increases from vo, thephase increases in magnitude but with negative sign (as occurs in conventional trans-mission lines). Regarding the behavior of wavelength for guided waves, lg, in thebalanced CRLH transmission line, this reaches its maximum (infinity) at vo, andlg decreases as frequency increases or decreases from vo. Indeed, for the unbalancedcase, the guided wavelength also increases when frequency approaches the spectralgap. However, at the edges of the gap the group velocity is zero, the line impedancetakes extreme values, and signal propagation is not allowed.

3.3 PRACTICAL IMPLEMENTATION OF BACKWARDTRANSMISSION LINES

As has been indicated in the previous section, the practical implementation of a dualtransmission line requires a host medium. As consequence, the description of thewhole structure bymeans of a lumped elementmodel includes not only the series capaci-tor and the shunt inductor, but also the elements of the host line, namely the per-sectioninductance and capacitance. Depending on the frequency region, propagation is back-ward (or left-handed) or forward (or right-handed) and the line behaves as a CRLHtransmission medium. Thus, rather than the implementation of backward (or purelyleft-handed) lines, which is not possible in practice, the main concern is the designand fabrication of CRLH transmission lines. These lines can be considered one-dimensional metamaterials belonging to the category of structures based on the dualtransmission line approach. Another type of one-dimensional metamaterial is thatbased on the resonant-type approach, where a host line is loaded with SRRs, or otherrelated topologies. The design of left-handed transmission lines based on SRRs willbe discussed in Section 3.5.

5Notice that by phase shift we refer to the phase of the transmission coefficient, which is of opposite sign tothe phase constant.


Returning to the CRLH lines, in principle, the host propagating medium can beany type of planar transmission line, such as microstrip, strip line or coplanar wave-guide (CPW) structures. For the reactive elements (series capacitor and shunt induc-tor) necessary to achieve a CRLH line, two possibilities arise, namely the use oflumped (typically surface mount technology) elements, or, alternatively, distributedcomponents. In this context, distributed components means planar elements thatare electrically small, that is, with a size smaller than lg. Typically, these planarelements are termed semi-lumped, because their size cannot be as small as thoseof lumped elements. However, the key point for the synthesis of metamaterials,that is, effectively homogeneous materials, is to make use of electrically small con-stitutive elements, or particles. Thus, it is important to reduce the size of the semi-lumped components as much as possible. The advantage of lumped elements istheir size; however, lumped elements can operate only in a limited frequency range(typically below 5–6 GHz) due to parasitic effects that cause self-resonance toappear. Moreover, it is difficult to achieve the required electrical characteristics byusing lumped elements, because their values are restricted to those provided by themanufacturers. Finally, the use of lumped elements is more expensive; soldering isnecessary (it causes additional losses) and it goes against the full integration of micro-wave components. Hence, semi-lumped elements are preferred.

Two main host lines have been considered for the synthesis of metamaterial trans-mission lines based on the dual (or backward) transmission line concept: the micro-strip structure and the coplanar waveguide configuration. The former was proposedby Caloz et al. in 2002 [9], and subsequently used in many applications. Theimplementation of a CRLH structure by means of a CPW configuration was due tothe group of Eleftheriades [10], who used the structure to demonstrate the occurrenceof backward-wave radiation in the fast-wave region of the dispersion diagram. Themicrostrip structure proposed by Caloz consists of a periodic arrangement of seriesinterdigital capacitors alternated with grounded (through metallic vias) stubs,which act as shunt-connected inductors (see Fig. 3.6). The interdigital capacitorsare described by the series capacitance CL in the circuit of Figure 3.3, whereas LLmodels the grounded stubs. The other reactive elements CR and LR correspond tothe line capacitance and inductance, respectively. As this book is mainly focusedon the resonant-type approach of metamaterials, we will not discuss the detailed pro-cedure to generate the layout, or to extract the electrical parameters from it [8].Nevertheless, we have included in Figure 3.7 the frequency responses (measured,simulated through electromagnetic solvers and by means of circuit simulation) of a

FIGURE 3.6 Microstrip CRLH transmission line implemented by means of shunt-connected grounded stubs and interdigital capacitors. (Source: Photo courtesy of C. Caloz.)

3.3 PRACTICAL IMPLEMENTATION OF BACKWARD TRANSMISSION LINES 129

FIGURE 3.7 Typical frequency responses for nine-cell balanced (a) and seven-cell unba-lanced (b) CRLH transmission lines of the type depicted in Figure 3.6. (Source: Reprintedwith permission from [8]; copyright 2006, John Wiley & Sons.)

FIGURE 3.8 CPW CRLH transmission line leaky wave antenna (a). The detail of the struc-ture and the radiation diagram are depicted in (b) and (c), respectively. (Source: Reprinted withpermission from [10]; copyright 2002, American Institute of Physics.)


nine-cell balanced structure and a seven-cell unbalanced design. The transitionfrequency is fo ¼ 3.9 GHz in the balanced structure, which does not exhibit thestopband. However, this stopband can be appreciated (separating the left-handedand right-handed frequency bands) in the unbalanced structure.

In the CPW configuration, the shunt inductors can be implemented by means ofconnecting strips between the central strip and ground planes. The series capacitorscan be implemented through interdigital geometries, or by means of series gaps.The latter are more simple, but the achievable capacitance values are muchsmaller. The Eleftheriades Group at the University of Toronto (Canada) presenteda realization of a left-handed line fabricated in CPW technology by using seriesgaps and shunt strips (see Fig. 3.8) [10]. This line was used as a leaky waveantenna, where the operating frequency was set in the fast-wave region of the spec-trum, where the structure may radiate. The most relevant aspect of this structure isthat, unlike ordinary leaky wave antennas, which exhibit forward radiation, thisline exhibits backward radiation. This effect has been explained as being due tothe propagation of backward leaky waves in the structure (see Section 1.8.3),and this can be considered as the signature of left-handed wave propagation in theradiating element. This aspect will be discussed in more detail in Chapter 4.

3.4 TWO-DIMENSIONAL (2D) PLANAR METAMATERIALS

In this section, the authors simply want to point out that the extension of the previoustransmission line, that is, planar one-dimensional (1D), metamaterials to 2D metama-terials is straightforward, rather than providing a network analysis of such 2D struc-tures. This has been done by others and is explained in [8] and [11]. Indeed, the first2D metamaterial was synthesized by the group of David Smith in 2001 by etchingSRRs and strips in multiple dielectric slabs orthogonally oriented [12] (seeFig. 2.26). Although the structure was utilized to demonstrate negative refraction, itwas a bulk structure, rather than planar. In planar technology, the first 2D structurewas proposed by the group of Eleftheriades [13], who experimentally demonstratedfor the first time the occurrence of near-field focusing [14]. The structure consistedof a 2D L–C loaded transmission line (see Fig. 3.9), where the capacitances wereimplemented by means of chip capacitors surface-mounted between gaps etched inthe grid lines between lattice nodes, and the inductances were realized throughchip inductors embedded into holes drilled at the node positions. The possibility ofovercoming the diffraction limit in a planar lens made of a 2D L–C loaded periodiclattice [15], which was placed between two 2D grids made of conventional microstriplines, was also experimentally demonstrated (Fig. 3.10). In Figure 3.10 are alsodepicted the measured vertical electric field detected in the structure (0.8mm abovethe surface), as well as the measured vertical electric field at the image, which is com-pared to the theoretical limit corresponding to a continuous medium. The amplifica-tion of evanescent waves and subdiffraction imaging are both demonstrated.

Alternatively, the possibility of implementing a 2D left-handed structure avoidingthe use of lumped elements has also been demonstrated [16]. The structure wasfirst proposed by Sievenpiper [17] for the realization of high-impedance surfaces.

3.4 TWO-DIMENSIONAL (2D) PLANAR METAMATERIALS 131

It consists of a mushroom structure (Fig. 3.11), which can be described by a CRLHtransmission line model in two dimensions (Fig. 3.12).6 The series capacitances (CL)are implemented by means of the slots present between neighboring patches, whereasvias are responsible for the shunt inductance (LL) necessary to obtain left-handed-ness. In the structure, the caps are used to enhance the weak capacitance couplingsbetween adjacent patches. The other elements of the circuit model (CR and LR) areprovided by the capacitance and inductance of the transmission line formed by theground plane and metal patches. The typical dispersion diagram for a periodicstructure based on the unit cell of Figure 3.12 is depicted in Figure 3.13 (seedetails for calculation and parameters in [16]) for the balanced and unbalancedcases. As occurs in the 1D case, backward and forward wave transmission arise indifferent frequency intervals, separated by a gap in the unbalanced case, and witha continuous transition under balance conditions. The dispersion diagram for thestructure of Figure 3.11 has been also obtained by Sanada et al. [16] from fullwave (FEM) simulation (Fig. 3.14). Similar characteristics as those of Figure 3.13are observed. However, the fundamental mode of the structure is mixed left-handed/right-handed, because, as argued in [16], the structure is opened and theleft-handed mode couples with the TM air mode (in a closed stripline structure apurely left-handed mode has been found [16]). On the other hand, there is also adegenerate right-handed TE mode not accounted for in the circuit model. The factthat the left-handed mode is shifted towards higher frequencies when the caps arenot present is indicative of the influence of the caps on the series capacitance CL.

Although this chapter is focused on planar structures, proposals of three-dimensional structures based on the transmission line approach of metamaterialshave also been done (see for instance [18] and [19]).

FIGURE 3.9 Two-dimensional metamaterial consisting of a grid of transmission linesloaded with series capacitors and shunt-connected inductors. (Source: Reprinted with per-mission from [13]; copyright 2002, IEEE.)

6This circuit model also describes the unit cell of those structures depicted in Figures 3.9 and 3.10.


FIGURE 3.10 Left-handed planar transmission line lens (a), measured vertical electric fielddetected 0.8 mm above the surface of the entire structure at 1.057 GHz (b), and measured ver-tical electric field at the image plane in solid line (c). For comparison, (c) also illustrates theelectric field at the source plane (dashed curve), and the theoretical diffraction-limitedpattern in a continuous medium (dash-dotted curve). (Source: Reprinted with permissionfrom [15]; copyright 2004, American Physical Society.)

3.4 TWO-DIMENSIONAL (2D) PLANAR METAMATERIALS 133

FIGURE 3.13 Typical dispersion diagram for a 2D periodic structure composed of the unitcells described by the circuit of Figure 3.12. (Source: Reprinted with permission from [16];copyright 2004, IEEE.)

FIGURE 3.11 General view of the 2D CRLH structure based on the mushroom configur-ation (a) and unit cell (b). (Source: Reprinted with permission from [16]; copyright 2004,IEEE.)

FIGURE 3.12 Equivalent circuit model of the unit cell of the structure shown in Figure 3.11.(Source: Reprinted with permission from [16]; copyright 2004, IEEE.)


3.5 DESIGN OF LEFT-HANDED TRANSMISSION LINES BYMEANS OF SRRs: THE RESONANT TYPE APPROACH

In the preceding sections of this chapter, it has been demonstrated that the dual trans-mission line concept can be used for the synthesis of planar left-handed transmissionlines in both microstrip and coplanar waveguide technologies. The negative effectivepermeability and permittivity values are provided by the series (capacitive) and shunt(inductive) impedances, respectively, loading the line. In fact, these impedances arenot purely capacitive and inductive, because the host transmission line introducesextra reactances of opposite sign (that is, series inductance and shunt capacitance)and their effects increase with frequency, with the result of a CRLH behavior thatlimits the bandwidth of the left-handed transmission band. Therefore, the backwardtransmission lines of the previous sections have a limited bandwidth, and theyexhibit forward transmission above the left-handed band.

The synthesis of left-handed transmission lines by properly etching SRRs in the hostline is also possible [20]. In contrast to the dual transmission line approach, we can termthis approach the resonant-type approach, because resonators (SRRs) are utilized asloading elements.7 The authorswould like to emphasize at this point that this designation(resonant-type approach) merely indicates that subwavelength resonators are used. Aswill be shown later, the transmission characteristics of both approaches (dual trans-mission line and resonant-type approach) are qualitatively very similar. In other

FIGURE 3.14 Typical dispersion diagram inferred from full wave (FEM) simulation andcorresponding to the structure depicted in Figure 3.11. (Source: Reprinted with permissionfrom [16]; copyright 2004, IEEE.)

7Alternatively, resonant-type metamaterial transmission lines can be implemented by using complementarysplit rings resonators (CSRRs). These particles will be introduced in Section 3.7.

3.5 DESIGN OF LEFT-HANDED TRANSMISSION LINES 135

words, resonant-type metamaterial transmission lines also exhibit a composite right-/left-handed (CRLH) behavior. Indeed, as for dual transmission lines, it is also possibleto design resonant-type balanced structures, with the absence of the stopband betweenthe left-handed and right-handed bands. Differences between the dual and resonant-type approach concern their descriptions through equivalent circuit models. Thesemodels are not identical, but they are similar. A comparison between both models andwith the transmission line model of bulk SRR-based metamaterials will be carried outlater in this chapter (Section 3.9).

The natural host lines for the implementation of SRR-based left-handed metamater-ials in one dimension are the microstrip line and the coplanar waveguide, among others.SRRs provide the negative effective permeability. However, to achieve backward-wavepropagation, further microstructuring is necessary, in order to obtain the required nega-tive effective permittivity. In coplanar waveguide technology, a possible solution is thatproposed by Eleftheriades and co-workers [10], namely, the periodic connection ofmetallic strips between the central strip and the ground planes. As has been discussedpreviously, these connecting wires act as shunt inductors and, hence, they provide anegative value of the effective permittivity up to a cutoff (or plasma) frequency thatdepends on the period of the structure, on the width of the shunt-connecting strips(which is intimately related to the inductance value) and on the per-unit cell capacitanceof the line. In microstrip technology, grounded stubs can be used to implement the nega-tive permittivity, as reported by the group of Professor Itoh at theUniversity of Californiaat Los Angeles (UCLA) [9]. Alternatively, the implementation of metallic vias betweenthe conductor strip and the ground plane at periodic positions arises as a possible solutionwhere lateral metallization (to etch the stubs) is avoided [21]. This is of interest, becauseSRRs must necessarily be etched in the upper substrate side in close proximity to theconductor strip to achieve high line-to-SRRs magnetic coupling. Therefore the lack ofstubs is preferred. Let us now consider in more detail the implementation of microstripandCPW transmission lines simply loaded with SRRs (that is, 1D negative permeabilitymedia). After this, the design of left-handed microstrip and CPW transmission lines willbe discussed in detail.

3.5.1 Effective Negative Permeability Transmission Lines

In order to load a CPW transmission linewith SRRs, and thus obtain a negative effectivepermeabilitymedium, it is necessary to etch the rings at the appropriate locations, that is,in such a way that the magnetic field generated by the current flowing through the lineexhibits a significant component in the axial direction of the rings. Under these con-ditions, the rings will be properly excited and the overall structure will be expected tobehave as an effective medium with negative-valued permeability in a certain bandabove the resonance frequency of the SRRs.8 This means that the SRRs must beetched in the slot region. However, they can be etched either at the upper metal level,or in the back substrate side (provided there is no ground metallization at that side of

8Although not explicitly indicated, by resonance frequency of the SRRs the authors refer to the first one(quasistatic resonance). This applies to the whole chapter.


the substrate). In the former case, wemay call the structure a uniplanar transmission line,but in the latter case it seems appropriate to talk in terms of a bi-metal implementation[22]. The uniplanar structure is more simple, as no back-side etching is required.However, it exhibits two drawbacks. First, wide slots are required to accommodatethe SRRs; this has a direct influence on the characteristic impedance of the hostline, which is expected to be high when compared to the usual impedance of 50Vrequired to obtain low insertion losses9 in the allowed frequency bands. On the otherhand, line-to-rings magnetic coupling is expected to be relatively weak provided themagnetic field flowing through the rings is only a few percent of the overall magneticflux generated by the host line (details on this structure including its behavior aregiven in [22]).

The preferred solution to obtain high line-to-rings coupling and low insertionlosses in the allowed band10 (that is, in that region where 1eff and meff are bothpositive) is the bimetal structure. Figure 3.15 depicts, the layout corresponding to abimetal implementation as well as the simulated and measured frequency responses.Because SRRs are etched in this case in the back substrate side, the lateral dimensionsof the host line (etched in the upper metal level) are no longer influenced by thepresence of the rings, and line matching is easily achieved. Moreover, as long asthe substrate is thin enough, the magnetic flux generated by the current flowingthrough the line can efficiently penetrate the SRRs and hence a high magnetic coup-ling between line and rings is expected. A stopband appears in the vicinity of fo ¼7.7GHz, the resonance frequency of the SRRs (their dimensions are indicated inthe caption of the figure and they have been calculated following [23] and Chapter2 to obtain such resonance frequency). In fact, the forbidden band extends aboveand below fo. This stopband can be interpreted as a consequence of the propertiesof the structure, which behaves as a 1D effective medium with negative magnetic per-meability in a narrow band above resonance and with high positive permeability in anarrow band below resonance,11 which causes a strong mismatch at the feeding portof the line. In Figure 3.15, it can be perfectly appreciated that no ripple is present inthe allowed band, which is indicative of a good matching. Indeed, outside the forbid-den band, the signal does not see the presence of the rings and it is propagatedbetween the input and output ports of the structure. A current diagram of the structurefor two different frequencies (that is, within the stopband and below) is depicted inFigure 3.16. At 2 GHz, the signal propagates and SRRs are not excited. However,at 7.7 GHz, SRR excitation is very visible and the injected power is retuned back

9In two-port networks it is usual to describe the transmission and reflection levels of the feeding signals bymeans of the insertion (IL) and return (RL) losses, respectively, where IL ¼ 20 logjS21j and RL ¼ 20logjS11j.10To efficiently use these negative permeability lines as small-sized stopband structures, it is necessary toreduce transmission band losses as small as possible. This application will be discussed in detail in the nextchapter.11Simultaneous positive or negative values of 1eff and meff are necessary but not sufficient conditions forsignal propagation. This is very clear to the light of Figure 3.2, where signal propagation is limited to acertain region of the electromagnetic spectrum.


to the source. It is interesting to notice that four ring pairs suffice to obtain a rejectionlevel in the vicinity of 40dB (Fig. 3.15).

In microstrip technology only one possibility arises, that is, the rings should beetched as close as possible to the line in order to obtain high magnetic coupling.In fact, in order to enhance this coupling, square-shaped or rectangular SRRs are pre-ferred in this case. This way, the magnetic flux lines penetrate the SRR’s area moreefficiently. Figure 3.17 depicts the layout and simulated frequency response of a 50Vmicrostrip transmission line loaded with square-shaped SRRs. Significant attenuationin the stopband is also achieved in this case, this band extending up and down theresonant frequency of SRRs.

FIGURE 3.15 Layout (a) and simulated (thin line) and measured (bold line) frequencyresponse (b) of the bi-metal SRR loaded CPW structure. SRRs, etched in the back side ofthe substrate, are depicted in black, whereas the upper metal level is depicted in gray. Ringdimensions have been determined following [23] to obtain a resonance frequency of fo ¼7.7 GHz, where ring width and separation are c ¼ d ¼ 0.2 mm and the radius of the innerring is r ¼ 1.3 mm. The distance l between adjacent rings is 5 mm. Lateral CPW dimensionshave been calculated to obtain a 50V characteristic impedance. The parameters used inthe simulation are those of the Arlon 250-LX-0193-43-11 substrate (1r ¼ 2.43, thicknessh ¼ 0.49 mm). (Source: Reprinted with permission from [22]; copyright 2004, John Wiley& Sons.)


3.5.2 Left-Handed Transmission Lines in Microstrip andCPW Technologies

The first left-handed transmission line implemented by means of SRRs was proposedby Martın et al. [20]. The structure was a coplanar waveguide with SRRs etched in theback substrate side, underneath the slots, and signal-to-ground metallic connections(strips) periodically located above the SRRs, as shown in Figure 3.18. In fact, thisstructure is identical to that shown in Figure 3.15, except for the presence of theshunt strips. These strips act as shunt-connected inductors, and hence they makethe structure behave as an effective medium with negative dielectric permittivity,up to a frequency (plasma or cutoff frequency, fc) that depends on the period andwidth of the shunt strips. Therefore, as long as fc is set above the resonance frequencyof the SRRs, a region exists where both the effective dielectric permittivity and mag-netic permeability (the latter provided by the rings) are both simultaneously negative.According to our previous words, this occurs in a narrow band above the resonancefrequency of the SRRs. Signal propagation in this band is thus allowed, but withbackward waves (that is, with negative phase velocity). The measured frequencyresponse of this structure is also depicted in Figure 3.18. This result confirms thatthe stopband behavior obtained when the shunt strips are not present switches to apassband in the SRR-loaded CPW with shunt strips. Moreover, there is a visible dis-placement towards positive frequencies in the passband, relative to the onset of the

FIGURE 3.16 Current diagrams for the bimetal structure of Figure 3.15, obtained by meansof CST Microwave Studio, at two different frequencies.


stopband in Figure 3.15. This is an expected result on account of the fact that thenegative effective permeability, necessary to obtain a band with negative wave propa-gation, arises at the resonant frequency of the SRRs.12 In contrast, the stopband in thenegative-permeability CPW (that is, without shunt-connecting strips) extends belowthe resonance frequency of the SRRs.

An interesting aspect of the structure shown in Figure 3.18 is the fact that the rela-tive position and orientation of the SRRs with regard to the shunt strips is relevant inorder to optimize the insertion losses in the allowed band. Namely, it has been foundthat by aligning the slits of the SRRs with the wire shorts, an optimum result in termsof in-band losses is obtained. An explanation of this is given in [24], and it has alsobeen given in Section 2.4.4, where the validity of the superposition principle in bulk

FIGURE 3.17 Layout (a) and simulated frequency response (b) of a 50V microstrip lineloaded with square-shaped SRRs. The structure has been implemented in the RogersRO3010 substrate (thickness h ¼ 1.27 mm, dielectric constant 1r ¼ 10.2). Dimensions: ringwidth and separation c ¼ 0.2 mm and d ¼ 0.4 mm, respectively, ring side (external) 5 mm,line width W ¼ 1.2 mm, and distance between adjacent rings 3 mm.

12In fact, the left-handed band begins slightly above the resonance frequency of the SRRs, as will be shownlater.


media composed of wires and SRRs was discussed. Essentially, thanks to this align-ment, no interference between the SRRs and the shunt strips takes place, because atthe SRR resonance there is a virtual electric wall along the orthogonal plane to therings containing the slits. Hence, if the relative orientation of the shunt strips andthe rings causes the strips to be contained in this imaginary plane, then such inter-action is minimized, this being a necessary condition for generating a 1D effectivemedium where both permeability and permittivity are negative.

In CPW technology, it is also possible to design left-handed structures with mul-tiple transmission bands. To this end, the buried CPW concept is invoked, where themetal level containing the ground planes and the central conductor strip is surroundedby a dielectric on top and bottom [24]. This allows SRRs to be etched in two separ-ated metal layers (on both sides of the substrate), which is of interest in tailoring thefrequency response of the structure, that is, to achieve two separated left-handedallowed bands (if the rings etched at opposite sides are tuned at different frequencies),or to widen the bandwidth (provided the SRRs are tuned at closer frequencies). Thisstructure can be considered as the parallel combination of two SRR-loaded CPW

FIGURE 3.18 Layout (a) and simulated (thin line) and measured (bold line) frequencyresponse (b) of the left-handed SRR-based line. SRR dimensions and separation as well aslateral dimensions of the CPW structure are identical to those of Figure 3.15. Wire width is0.2 mm. (Source: Reprinted with permission from [20]; copyright 2003, American Instituteof Physics.)


transmission media, where the host CPW is shared. This explains the presence of twotransmission bands, which can be partially overlapped to produce a single trans-mission band with improved bandwidth. To illustrate these ideas, two buried CPWstructures are provided, one with the top and bottom SRRs tuned at closed frequencies(Fig. 3.19), and the other designed to provide separated narrow pass-bands(Fig. 3.20). The simulated and measured frequency responses for these structuresare also depicted in Figures 3.19 and 3.20. The slight frequency shift between simu-lations and experiments is due to some discrepancy between the dimensions of thefabricated rings and the nominal values. This may also cause the degradation of

FIGURE 3.19 Layout (a) and frequency response (b) of the left-handed buried CPW struc-ture with closely tuned SRRs. Measured results are depicted as bold lines, and simulations aredepicted as thin lines. To better distinguish, measured return losses have been depicted in gray.Geometrical parameters are c ¼ d ¼ 0.2 mm, inner radius for the top and bottom ringsr ¼1.9 mm and r ¼ 2.0 mm, respectively (the parameters of the Arlon 250-LX0193-43-11 sub-strate have been considered: 1r ¼ 2.43, thickness h ¼ 0.49 mm). With these dimensions, thenominal resonant frequencies for the top/bottom rings are in the vicinity of 5 GHz. Thewidth of connecting wires (0.2 mm) guarantees a plasma frequency beyond the resonant fre-quency of SRRs. Slot and strip widths (G ¼ 0.3 mm, W ¼ 5.5 mm) correspond to a 50Vline. Simulations have been carried out using CST Microwave Studio. (Source: Reprintedwith permission from [24]; copyright 2004, IEEE.)


the second passband for the structure of Figure 3.20, which is too close to the plasmafrequency of the CPW structure loaded with shunt strips. Nevertheless, the presenceof two allowed bands confirms that the device can be viewed as two separated left-handed structures parallel connected, and it explains bandwidth widening when thetwo sets of rings are closely tuned. The high-frequency selectivity and low level oflosses (allowed band) measured in both structures (,1.5 dB) is remarkable.

In microstrip technology, it has previously been anticipated that the combinationof SRRs (etched as close as possible to the conductor strip to enhance magnetic coup-ling) and via holes is appropriate for the synthesis of left-handed structures in onedimension [21]. The metallic vias emulate shunt-connected inductors and providethe required negative permittivity, whereas SRRs are responsible for the negativeeffective permeability. In Figure 3.21, it is depicted a left-handed line with theseelements (the measured and simulated frequency responses are also depicted).Again, as expected, a passband behavior has been obtained, the allowed band extend-ing over a narrow region above the resonance frequency of the SRRs. In this case thegeometry of the rings has been chosen to be square-shaped to enhance magneticcoupling between the line and the rings, as has previously been commented.

FIGURE 3.20 Layout (a) and frequency response (b) of the left-handed buried CPW struc-ture with two separated transmission bands. Measured results are depicted as bold lines, andsimulations are depicted as thin lines. To better distinguish, measured return losses havebeen depicted in gray. The inner radius of SRRs etched in the top metal level has been setto r ¼ 1.2 mm, and other geometric parameters are identical to those given in Figure 3.19.(Source: Reprinted with permission from [24]; copyright 2004, IEEE.)


3.5.3 Size Reduction

In the previous implementations of metamaterial transmission lines based on SRRs,dimensions were relatively small on account of the small electrical size of the SRRs.For instance, in the left-handed CPW structure shown in Figure 3.18, the length, exclud-ing access lines, is as small as l � l/2, l being the wavelength corresponding to a 50Vline implemented onto a identical substrate at SRR resonance.13 To reduce the size ofSRR-based structures, it is necessary to decrease the electrical size of SRRs, and tothis end it is imperative to reduce the space between the rings and/or to narrow theirwidth. The effect is a shift of the resonance frequency of the SRRs to lower valuesand hence an improvement of their electrical size. However, in practice, due to thelimited lateral resolution of standard fabrication systems (typically in the vicinity of100mm for chemical etching or drilling machines on conventional microwave

FIGURE 3.21 Layout (a) and simulated (dashed line) and measured (solid line) (b) fre-quency response for the microstrip left-handed line implemented by means of square-shapedSRRs and metallic vias. Substrate and dimensions are as in Figure 3.17. (Source: Reprintedwith permission from [21]; copyright 2005, IEEE.)

13Notice that l= lg (see Section 3.2).


FIGURE 3.22 Layout (a), simulated (b), and measured (c) frequency response of thenegative-permeability CPW structure loaded with SRs. SRs, etched in the back side of the sub-strate, are depicted in black, and the upper metal level is depicted in gray. SR dimensions andseparation are roughly the same as those of Figure 3.15. Lateral CPW dimensions have beencalculated to obtain a 50-V characteristic impedance. (Source: Reprinted with permissionfrom [27]; copyright 2004, John Wiley & Sons.)


substrates), it is difficult to drive SRRdimensions (external diameter) below one-tenth ofa wavelength at resonance (this being indeed an optimistic prospect).

To further reduce the size of the devices and circuits, alternative topologies for theresonators may be more convenient. As has already been pointed out in Chapter 2(Section 2.3), the BC-SRR, where one ring is placed above the other, allows forhigher levels of miniaturization [25]. This is possible because, thanks to the broadside coupling (which can be enhanced by simply reducing the width of the inter-ring dielectric), the equivalent capacitance of the structure is increased. Another possi-bility is to replace the SRRs with spiral resonators (SRs) [26,27]. By etching them ontothe same substrate, the electrical size of a two-turn SR is roughly half the electrical sizeof an SRR, provided dimensions are identical. To illustrate this, Figure 3.22 depicts anegative-permeability CPW transmission line loaded with two-turn SRs, together withthe simulated and the measured frequency responses. The dimensions of the SRs arevery similar to those of the SRRs of Figure 3.15. Again, a stopband behavior related tothe negative effective permeability provided by the SRs is obtained. However, the for-bidden band arises in the vicinity of 3.5GHz, which is approximately half the fre-quency of the central frequency for the forbidden band depicted in Figure 3.15.Obviously, by increasing the number of turns of the spirals, it is possible to furtherdecrease their electrical size. However, this makes ohmic losses significant for practi-cal applications, unless high-temperature superconductors (HTS) are used [28]. It isworth mentioning that the combination of SRs and shunt strips to achieve a left-handed transmission band has been considered by the authors, but poor frequencyresponses as compared to those of Figures 3.18–3.21 have been obtained. It is believedthat the reason for this degradation is the lack of symmetry of the SR with regard to theplane of the shunt strips, which may cause a nonnegligible interaction between SRsand strips.

3.6 EQUIVALENT CIRCUIT MODELS FOR SRRs COUPLED TOCONVENTIONAL TRANSMISSION LINES

In the previous chapter, it was demonstrated that SRRs can be modeled by an LC res-onant tank that can be externally driven by a time-varying magnetic field applied inthe axial direction. The equivalent capacitance is the series connection of the edgecapacitance corresponding to the lower and upper halves of the structure, and theequivalent inductance is given by the inductance of a closed ring with the samewidth of each individual ring and average SRR radius. For the alternative topologiesof the SRR presented in Chapter 2, similar equivalent circuits arise. All of them canbe reduced to a simple parallel LC tank, as for the SRRs. However, in order to designmetamaterial transmission lines based on SRRs, equivalent circuit models for theselines are necessary. These models should describe the host transmission line, the reso-nators (SRR or other resonators), and their coupling. It was indicated in the previouschapter that the basic SRR topology exhibits cross-polarization effects. This meansthat SRRs can be magnetically and/or electrically excited if the rings are properlyoriented. However, it has been verified that magnetic coupling is the dominant


coupling mechanisms in SRRs. Therefore cross-polarization effects can be ignored ina first-order approximation (this assumption is strictly valid for NB-SRRs and othernonbianisotropic configurations under a uniform field excitation).

Due to the small electrical dimensions of SRRs at resonance, the structures (CPWor microstrip loaded lines) can be described by means of lumped element equivalentcircuits. The proposed equivalent circuit model (basic cell) for the SRR loaded trans-mission line (of Figs 3.15 and 3.17) is shown in Figure 3.23a [22]. L and C are theper-section inductance and capacitance of the line, and SRRs are modeled as resonanttanks (with inductance Ls and capacitance Cs) magnetically coupled to the linethrough a mutual inductance,14 M. For the SRR-loaded CPW, this mutual inductancecan be inferred from the fraction, f, of the slot area occupied by the rings, according to

M ¼ 2L � f , (3:34)

whereas for the microstrip structure it is difficult to accurately determine M and it isconsidered a fitting parameter. Neither losses (ohmic and dielectric) nor inter-resonator coupling have been considered in this model.15 The equivalent impedance

FIGURE 3.23 Lumped element equivalent p-circuit model of the SRR loaded negative-permeability transmission lines (a), and circuit model that results after transformation of theseries branch (b).

3.6 EQUIVALENT CIRCUIT MODELS FOR SRRs 147

14In the circuit models of Figure 3.23, the presence of a virtual magnetic wall at the middle plane of thelines of Figures 3.15 and 3.17 has been taken into account, so that only one of the SRRs loading eachsection of the line is included in the model.15Coupling between adjacent SRRs is only significant in closely spaced square or rectangular SRRs. Suchcoupling is the origin of magnetoinductive waves in chains of SRRs, and it will be considered in the lastchapter. However, it can be neglected in modeling the structures considered in this chapter.

of the series branch can be simplified to that shown in the circuit of Figure 3.23b,which is formally identical to the series impedance corresponding to a left-handedtransmission line (in that region where the total series impedance is capacitive). Todemonstrate this, the voltage–current equations for the transformer are first derived(in this analysis, to gain further insight on the effect of losses, a series resistance,Rs, will be included in the LC tank modeling the SRRs). These equations are

V1(v) ¼ jv2LI1(v)þ jvMI2(v) (3:35)

and

�I2(v)Za(v) ¼ jvMI1(v)þ jvLsI2(v), (3:36)

where the variables refer to the circuit of Figure 3.23a, and Za is the series connectionof the equivalent capacitance of the SRR, Cs, and Rs, namely:

Za(v) ¼1þ jvRsCs

jvCs: (3:37)

By isolating I2 in equation (3.36), and substituting it into equation (3.35), we directlyobtain the series impedance of the line:

Zs(v) ¼ j2vLþ v2M2 jvCs

(1� v2LsCs)þ jvRsCs

� �: (3:38)

This impedance is equivalent to the series connection of an inductor (with inductance2L) and a parallel RLC tank, provided the following conditions are satisfied:

L0s ¼ v2oM

2Cs (3:39)

C0s ¼ Ls=v

2oM

2 (3:40)

R0 ¼ v2oM

2=Rs, (3:41)

where vo ¼ (CsLs)�1=2 ¼ (C0sL

0s)�1=2 is the resonance frequency of the SRRs.

Therefore, the equivalent circuit model of Figure 3.23a can be simplified to thatshown in Figure 3.23b (where R0 has not been included for coherence).

For the left-handed structures shown in Figures 3.18 and 3.21, where either ashunt-connected strip or a metallic via has been introduced (compared to thenegative-permeability structures), the equivalent circuit model is identical to thatshown in Figure 3.23a (or to the simplified version of Fig. 3.23b), but with theaddition of a shunt-connected inductance that models the signal to ground strips(CPW configuration) or the vias (microstrip structures). This circuit is depicted inFigure 3.24. The value of the shunt inductance, Lp, can be inferred from the simulatedfrequency response of the host line with SRRs removed, where the plasma frequencyis given by the resonator composed by C and Lp [20].


A comparison between the frequency responses measured on the CPW meta-material structures and those obtained through electrical simulation of the equivalentcircuit models (with Ls and Cs obtained from the expressions in the Appendix ofChapter 2) is depicted in Figure 3.25 [29]. Reasonable agreement between modeland experiment results, if one takes into account that losses are neglected and theelectrical parameters of the equivalent circuit model are all estimated, rather thanused as fitting parameters.

It is worth mentioning that the model of Figure 3.24 also predicts a forward-wavetransmission band above the left-handed region (CRLH behavior). As for the trans-mission lines loaded with series capacitances and shunt inductances studied inSection 3.2, at high enough frequencies the parameters of the host line become domi-nant and signal propagation is forward. The shunt impedance changes its sign at theresonance frequency of the tank formed by Lp and C:

vp ¼1ffiffiffiffiffiffiffiffiCLp

p : (3:42)

The series reactance is negative in the left-handed band and it switches to a positivevalue (that is, inductive) at the following angular frequency:

vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2C0s L

þ 1L0s C

0s

s: (3:43)

FIGURE 3.24 Lumped element equivalent p-circuit model of the SRR loaded left-handed transmission lines (a), and circuit model that results after transformation of theseries branch (b).


Thus, the limits of the stopband present between the left-handed and right-handedregions are given by expressions (3.27) and (3.28), with vp and vs given byexpressions (3.42) (a) and (3.43) (b). Usually, these left-handed lines based onSRRs are designed with vp . vs. In other words, the cutoff (plasma) frequencyrelated to the presence of the shunt strips or vias is set significantly above the reson-ance frequency of the SRRs (which is close to vs in practical implementations).The special case where vp ¼ vs collapses the gap and provides the balancedSRR-loaded left-handed transmission line.16

FIGURE 3.25 Comparison between the measured insertion losses (solid line) and thoseinferred from the equivalent circuit models (dashed line) for the negative permeability structureof Figure 3.15 (a) and for the left-handed structure of Fig. 3.18 (b). (Source: Reprinted withpermission from [29]; copyright 2005, IEEE.)

16Practical realizations of balanced CRLH resonant-type metamaterial transmission lines implemented byusing complementary split rings resonators (CSRRs) will be presented later in this chapter.


In the structures presented in Figures 3.18–3.21, the interest was to achieve abackward wave band. Nevertheless, the gap and the right-handed band above itcan be clearly appreciated in Figures 3.19 and 3.20 (in Figs 3.18 and 3.21 the hori-zontal scale is limited to show only the backward transmission band).

3.6.1 Dispersion Diagrams

For the equivalent p-circuit models obtained for the negative permeability and left-handed transmission lines based on SRRs or other related topologies, the dispersiondiagram is also given by expression (3.1). Zs(v) and Zp(v) are the series and shuntimpedances of the p-circuit models. By applying (3.1) to the circuits shown inFigures 3.23 and 3.24, we obtain

cos(bl) ¼ 1� LCv2

2þ C=C0

s

4 1� v2o

v2

� � (3:44)

cos(bl) ¼ 1� 12LCv2 1�

v2p

v2

!1� 1

2LC0sv

2 1� v2o

v2

� �0BB@

1CCA, (3:45)

with vp given by equation (3.42). These expressions have been evaluated by using theelectrical parameters corresponding to the structures depicted in Figures 3.15 and3.18, and they are represented in a reduced Brillouin diagram in Figure 3.26. For com-parison purposes, in this figure there are also represented the dispersion diagrams thathave been obtained from the S-parameters computed by electromagnetic simulationof a single unit cell, according to the following expression [6]:

cosf ¼ Aþ D

2, (3:46)

where A and D are the diagonal elements of the ABCD matrix, which are related toS-parameters through well-known expressions [30]. As expected, a stopband appearsin the vicinity of the resonant frequency of the SRRs for the negative permeabilitystructure, whereas this stopband is switched to a bandpass behavior with antiparallelphase and group velocities for the left-handed structure.17

3.6.2 Implications of the Model

Let us now analyze in more detail the transmission properties of SRR-based left-handed transmission lines. These structures are by nature frequency-selective


17The portion of the dispersion diagram of the left-handed line corresponding to the right-handed band hasnot been represented in the figure.

devices that exhibit a bandpass behavior. In Chapter 4, the use of SRR-based struc-tures for the implementation of planar microwave filters will be discussed exhaus-tively, and it will be shown that it is possible to design compact and high-performance filters with controllable characteristics. However, before this discussion,it is necessary to point out the main relevant characteristics of metamaterialtransmission lines based on SRRs. This will provide us with interesting information,which will be helpful in order to predict the potentiality of SRR-based devices asfiltering structures and to envisage other possible circuit applications. For theanalysis that we want to carry out, the frequency dependence of the Bloch impedanceof the structure will provide us with valuable information. In reference to the p-circuit model of Figure 3.24b, the Bloch impedance is given by the followingexpression [6]:

ZB(v) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZs(v)Zp(v)=2

1þ Zs(v)2Zp(v)

vuuut , (3:47)

where Zs and Zp are the series and shunt impedances of the p-circuit, respectively.The allowed band for backward-wave propagation in the structures occurs in thatregion where the Bloch impedance, ZB, and the phase constant, f ¼ bl, given byexpression (3.1) are both real numbers. From equation (3.1), the limits of this interval

FIGURE 3.26 Theoretical (solid lines) and simulated (dashed lines) dispersion diagram forthe infinite periodic SRR-CPW structures with unit cells identical to those shown in Figures3.15 and 3.18. (Source: Reprinted with permission from [29]; copyright 2005, IEEE.)


are given by the following conditions:18

Zs(v) ¼ 0 (3:48)

where f ¼ 0 and ZB ¼ 0, and

Zs(v) ¼ �2Zp(v), (3:49)

where f ¼ 2p and ZB ¼ 1. Expression (3.48) leads to the upper limit of the left-handed band, namely

fH ¼ 12p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2C0sL

þ 1L0sC

0s

s, (3:50)

and the lower limit can be inferred from equation (3.49):

fL ¼ 12p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

C0s(2Lþ 8Lp)

þ 1L0sC

0s

s: (3:51)

To obtain expression (3.51), it has been assumed that the shunt reactance is domi-nated by Lp, that is, C has been neglected. This approximation is reasonable aslong as vp � vs, and it simplifies expression (3.51). In the interval delimited bythe previous frequencies ( fL and fH), the sign of the series, Zs(v), and shunt,Zp(v), impedances is negative and positive, respectively. With the signs of these reac-tances, the structure formally behaves as a dual transmission line; that is, it admitsbackward waves. It is interesting to mention at this point that, as was previouslyanticipated, the onset of left-handed wave propagation does not exactly coincidewith the resonance frequency of the SRRs, fo, despite the fact that these resonantelements provide the negative value of meff just above this frequency. Indeed, at fothe series branch of the p-circuit model opens and the signal is completely reflected(transmission zero).19

According to equations (3.50) and (3.51), the left-handed bandwidth depends onthe values of the electrical parameters of the equivalent circuit model. As low as L iscompared to Lp and L0s, the wider the bandwidth can be. However, this is not the con-ventional 3dB bandwidth of filters, delimited by those frequencies where insertionlosses drop to 3dB. The left-handed bandwidth, delimited by fL and fH, must beunderstood as that region of the spectrum where the phase constant, b, is real and


18At that frequency where Zp(v)¼ 1 (expression 3.42) the phase shift per cell is also f ¼ 0, whereas ZB ¼

1. This frequency is the lower limit of the forward wave band. Above this frequency, the signs of the seriesand shunt reactance are positive and negative, respectively.19Under ideal conditions (lossless line and SRRs).

negative, regardless of the transmission coefficient. In fact, at fL and fH the impedancemismatch between the source, the load, and the line is maximum, because the Blochimpedance takes extreme values at these frequencies. Thus the 3dB bandwidth islower than the left-handed bandwidth. However, the frequency response within theleft-handed band is not only controlled by impedance matching, but also by phasematching. In other words, total transmission not only occurs when the Bloch impe-dance coincides with the reference impedance of the ports20 (usually 50V), butalso when the electrical length of the structure is a multiple of p. The latter condition(phase-matching) takes place at N 2 1 different frequencies (N being the number ofdevice stages) satisfying the following rule (relative to expression 3.45):

fi ¼p

Ni (3:52)

where 0, i , N. Hence, the number of transmission peaks equals the number ofbasic cells,21 N. As N increases, the transmission peaks spread out towards thelimits given by expressions (3.50) and (3.51). The effect is an enhancement of the3dB bandwidth. By using the electrical parameters corresponding to the modelthat describes the left-handed microstrip line of Figure 3.21, the electrical responsehas been obtained (Fig. 3.27). The transmission peaks are clearly visible, althoughin measurement these peaks are obscured by losses, and they do not emerge (seeFig. 3.21). In practice, as a first-order approximation and provided that the numberof cells is not very small, it can be assumed that the left-handed and 3dB bandwidthscoincide.

We have mentioned in the preceding paragraph that the left-handed bandwidthdepends on the relative value of L as compared to Lp and L0s. In practice, the line

FIGURE 3.27 Electrical response of the circuit of Figure 3.24, with the parameters corre-sponding to the four-stage left-handed structure of Figure 3.21.

20This occurs at a single frequency.21Unless one of the frequencies given by expression (3.52) coincides with that frequency where impedancematching is satisfied.


inductance cannot be driven to very low values (as this would require very wide con-ductor strips). On the other hand, the inductance of the shunt metallic strips or viascannot be increased too much, because the plasma frequency, determined by the reso-nator formed by this inductance and the line capacitance, must remain beyond theSRR resonance. For these reasons, it is not simple in practice to design wide-bandleft-handed transmission lines by combining SRRs and shunt-connected strips (orvias). However, the reason, rather than being related to the quality factor of theSRRs, is due to the difficulty of satisfying the requirements for wide allowablebands by using the structures under study. In Chapter 4, it will be shown that, byintroducing additional elements to the structures, it is possible to achieve verywide-band left-handed transmission lines and that these 1D metamaterials can beapplied to the design of planar microwave filters with controllable characteristics.

A relevant characteristic of the left-handed transmission lines based on SRRs is thepresence of a transmission zero at the resonance frequency of SRRs (as has been pre-viously indicated). This transmission zero is not present in the LC loaded dual (orCRLH) structures22 considered in Section 3.2. Due to this transmission zero, thelower band edge of the left-handed band can be made sharp. To this end, Lp must beincreased, although this degrades frequency selectivity in the upper edge of the band,because the cutoff (plasma) frequency related to the presence of the vias decreases.Nevertheless, for reasonable values of the electrical parameters of the model, that is,for implementable device geometries, typically the lower band edge is sharp, whereasfrequency selectivity at the upper band is smooth, as can be appreciated inFigure 3.21. In Chapter 4, it will be shown that by alternating SRR/strip (or via)stages with basic cells consisting of the combination of SRRs with capacitive gaps(that is, right-handed stages), it is possible to improve frequency selectivity at theupper band edge.

To end this section, we would like to mention that, apart from the frequency-selective properties of SRR-based devices (which are of interest for microwave filterdesign), it has been shown that the Bloch impedance and the phase shift per cellvary over a wide margin within the allowed band. This points out the potential ofthese structures in those applications where the control of impedance and phase is ofinterest, as occurs in most microwave circuits. As will be studied in the next chapterof this book, many applications, where miniaturization is a key aspect, will bederived from this fact.

3.7 DUALITY AND COMPLEMENTARY SPLIT RINGRESONATORS (CSRRs)

In this section, the concept of complementary split ring resonators (CSRRs) is intro-duced as an alternative to the design of metamaterials based on resonant elements,providing an effective negative permittivity, rather than permeability [31]. By invok-ing the concepts of duality and complementarity [32–34], the CSRR can be derived

22In fact, the transmission zero for LC loaded lines is at the origin.

3.7 DUALITY AND COMPLEMENTARY SPLIT RING RESONATORS 155

from the SRR structure in a straightforward way. This particle, which in planar tech-nology can be defined as the negative image of the SRR, exhibits an electromagneticbehavior that is almost the dual of that of the SRR. Specifically, a negative effectivepermittivity can be expected for any CSRR-based medium, whereas a negative-mbehavior arises in an equivalent SRR-based system.

3.7.1 Electromagnetic Properties of CSRRs

The electromagnetic properties of SRRs have been already analyzed in Chapter 2.This analysis shows that SRRs behave as an LC resonator that can be excited byan external magnetic flux, exhibiting a strong diamagnetism above their first reson-ance. SRRs also exhibit cross-polarization effects (magnetoelectric coupling) [25],so that excitation by a properly polarized time-varying external electric field is alsopossible. Figure 3.28 again reproduces the basic topology of the SRR, as well asthe equivalent circuit model, for comparison purposes. In this diagram, Co indicatesthe total capacitance between the rings, that is, Co ¼ 2proCpul, where Cpul is the per-unit-length edge capacitance. The resonance frequency of the SRR is given by fo ¼(LsCs)

21/2/2p, where Cs is the series capacitance of the upper and lower halves of theSRR, that is, Cs ¼ Co/4. The inductance, Ls, can be approximated by that of a singlering with averaged radius ro and width c [25,29].

As is well known, the complementary of a planar metallic structure is obtained byreplacing the metal parts of the original structure with apertures, and the apertures

FIGURE 3.28 Topologies of the SRR (a) and CSRR (b), and their equivalent circuit models(ohmic losses can be taken into account by including a series resistance in the model). Grayzones represent the metallization. (Source: Reprinted with permission from [29]; copyright2005, IEEE.)


with metal plates [32]. Due to symmetry considerations, it can be demonstrated that, ifthe thickness of the metal plate is zero, and its conductivity is infinity (perfect electricconductor), then the apertures behave as perfect magnetic conductors. In that case theoriginal structure and its complementary are effectively dual, and if the fieldF ¼ (E,H )is a solution for the original structure, its dual F 0 defined by

F0 ¼ (E0,H0) ¼ (�ffiffiffiffiffiffiffiffiffim=1

p� H,

ffiffiffiffiffiffiffiffiffi1=m

p� E), (3:53)

is the solution for the complementary structure [33]. Thus, under these ideal con-ditions, a perfectly dual behavior is expected for the complementary screen of theSRR. Thus, whereas the SRR can be mainly considered as a resonant magneticdipole that can be excited by an axial magnetic field, the CSRR (also depicted inFig. 3.28) essentially behaves as an electric dipole (with the same frequency of res-onance) that can be excited by an axial electric field. In a more rigorous analysis, thecross-polarization effects in the SRR [25], discussed in Chapter 2, should be con-sidered and extended also to the CSRR. Thus, this last element will also exhibit aresonant magnetic polarizability along its y-axis (see Fig. 3.28) and, therefore, itsmain resonance can be also excited by an external magnetic field applied alongthis direction [29].

Rigorously speaking, due to the continuity of the tangential/normal componentsof the electric/magnetic fields in the CSRR plane, F0 in equation (3.53) is the solutionon one side of this plane, and 2F0 on the other side. With regard to the CSRR polar-izabilities, this means that the sign of such polarizabilities changes from one side toanother. Therefore, the net electric and magnetic dipoles on the CSRR must vanish, aresult that can be also deduced from the fact that electric currents confined in aplane cannot produce any net normal/tangential electric/magnetic polarization.However, when the CSRR is seen from one side, the aforementioned effectivepolarizabilities arise.

The intrinsic circuit model for the CSRR (dual of the SRR model) is shown inFigure 3.28. In this circuit [29], the inductance Ls of the SRR model is substitutedby the capacitance, Cc, of a disk of radius ro 2 c/2 surrounded by a ground planeat a distance c of its edge. Conversely, the series connection of the two capacitancesCo/2 in the SRR model is substituted by the parallel combination of the two induc-tances connecting the inner disk to the ground. Each inductance is given by Lo/2,where Lo ¼ 2proLpul and Lpul is the per-unit-length inductance of the CPWs connect-ing the inner disk to the ground. For infinitely thin perfect conducting screens, and inthe absence of any dielectric substrate, it directly follows from duality that the par-ameters of the circuit models for the SRRs and the CSRRs are related by Cc ¼

4(1o/mo)Ls and Co ¼ 4(1o/mo)Lo. The factor of 4 appearing in these relations isdeduced from the different symmetry properties of the electric and magnetic fieldsof both elements, as is sketched in Figure 3.29. From the above relations it iseasily deduced that the frequency of resonance of both structures is the same, as isexpected from duality.

It has already been mentioned that the behavior of SRRs and CSRRs is strictlydual for perfectly conducting and infinitely thin metallic screens placed invacuum. However, deviations from duality—which may give rise to a shift in


the frequencies of resonance—arise from losses, the finite width of metallizations,and the presence of a dielectric substrate. The latter is expected to be the maincause of deviations from duality. This fact is due to the variations of the elementsof the CSRR circuit model, Cc and Lo, from the values extracted from the SRRcircuit model parameters Co and Ls by duality (Cc ¼ 4(1o/mo)Ls and Co ¼ 4(1o/mo)Lo). As is sketched in Figure 3.30, these variations arise directly from the pre-sence of a dielectric substrate, which affects Cc and Co but leaves Ls and Lo unal-tered. Analytical expressions for Ls and Co in the SRR when a dielectric substrateis present were provided in Chapter 2 [23]. As we have already mentioned, thecapacitance Cc in Figure 3.29 is that corresponding to a metallic disk of radiusro 2 c/2 surrounded by a ground plane at a distance c. An analytical approximateexpression for Cc when a dielectric substrate is present (see Fig. 3.31) has beenderived in reference [29]. The final expression is

Cc ¼p310

c2

ðþ1

0dk

bB(kb)� aB(ka)½ �2

k212

1þ1þ 1

10tan h(kh)

1þ 10

1tan h(kh)

0B@

1CA

264

375, (3:54)

FIGURE 3.29 Sketch of the electric and magnetic field lines in the SRR (left) and theCSRR (right). (a) Electric field lines in the SRR at resonance. (b) Magnetic field lines in thedual CSRR. (c, d ) Magnetic and electric field lines in the SRR and the CSRR, respectively.(e) Magnetic induction field in the equivalent ring inductance used for the computation ofLs in the SRR [31]. ( f ) Electric field in the dual equivalent capacitor proposed for the compu-tation of Cc for the CSRR. (Source: Reprinted with permission from [29]; copyright 2005,IEEE.)


where a, b, and h are geometrical variables defined in Fig. 3.31 and function B isdefined as

B(x) ¼ S0(x)J1(x)� S1(x)J0(x), (3:55)

with Sn and Jn being the nth-order Struve and Bessel functions. The inductance Lo inFig. 3.28 is that corresponding to a circular CPW structure of length 2pro, strip widthd, and slot width c. The design formulas given in [35] for the per-unit-length CPWinductance provide enough accuracy and have been used in all numerical compu-tations of the next subsection.

The previous analysis can be easily extended to the complementary version ofother planar topologies derived from the basic geometry of the SRR, such as theNB-SRR, the 2-SRR (double-slit SRR) and the SR. The equivalent circuit modelsfor such complementary particles are depicted in Figure 3.32, jointly with the

FIGURE 3.30 Sketch of the electric (a) and magnetic (c) field lines of an SRR on a dielectricsubstrate. The magnetic (b) and electric field (d ) lines of a similar CSRR on the same dielectricsubstrate are also sketched. (Source: Reprinted with permission from [29]; copyright 2005,IEEE.)

FIGURE 3.31 The capacitance of the CSRR is approximately equal to that corresponding toa metallic disk of radius a ¼ ro 2 c/2 surrounded by a ground plane at a distance b 2 a ¼ c, robeing the averaged radius of the CSRR and c the width of the slots in this. The dielectric sub-strate is characterized by its permittivity 1 and thickness h. (Source: Reprinted with permissionfrom [29]; copyright 2005, IEEE.)


equivalent circuits of the noncomplementary counterparts, which are reproducedhere for comparison purposes.

3.7.2 Numerical Calculation and Experimental Validation

The effect of the dielectric substrate on the relative shift between the resonance fre-quency of CSRRs and SRRs of identical dimensions is shown in Figure 3.33. Asexpected, there is no difference for the two limiting values of a zero- and an infinitesubstrate thickness. However, significant differences in the values of the frequency of

FIGURE 3.32 Topologies corresponding to (a) the nonbianisotropic SRR (NB-SRR), (b)the double-slit SRR (2-SRR), (c) the spiral resonator (SR), and (d ) the double SR(DSR).The equivalent circuits for these topologies are depicted in the second column, andthe circuit models for the complementary counterparts are represented in the third column.(Source: Reprinted with permission from [29]; copyright 2005, IEEE.)


resonance for both elements can be observed for intermediate thickness. Obviously,these differences increase with the dielectric constant of the substrate (see alsoFig. 3.33). In order to experimentally verify the accuracy of the proposed circuitmodels for the CSRR and derived geometries, a set of these resonators with differenttopologies were etched on a metallized microwave substrate and its frequencies ofresonance were measured. Their dual counterparts were also manufactured andmeasured for completeness. The resonance frequencies were obtained from the trans-mission coefficient, S21, measured in a rectangular waveguide, properly loaded withthe corresponding element [29]. The waveguide was excited in the fundamental TE01

mode, and connected to an Agilent 8510 network analyzer. The SRRs or derived geo-metries were placed in the central E-plane, so that they were excited by the magneticfield perpendicular to the element plane. Their dual counterparts were etched in thetop wall of the waveguide, being excited by the electric field perpendicular to the

FIGURE 3.33 Numerical calculations showing the dependence of the resonance frequencyof SRRs (solid lines) and CSRRs (dashed lines) on the substrate parameters. (a) Dependenceon the dielectric thickness for different values of the relative permittivity of the substrate(shown at right). (b) Dependence on the value of the relative dielectric constant, for differentsubstrate thickness (in mm). (Source: Reprinted with permission from [29]; copyright 2005,IEEE.)


element plane. Figure 3.34 shows the transmission coefficients for an SRR and anNB-SRR with identical geometrical parameters, as well as the same coefficientsfor their duals (CSRR and C-NB-SRR). The SRR and the NB-SRR have the samefrequency of resonance (the small shift can be attributed to tolerances in the manu-facturing process). The same can be said for their complementary elements.Finally, the frequencies of resonance for different configurations, measured by fol-lowing the method illustrated in Figure 3.34, are shown in Table 3.1 [29]. The theor-etical values shown in this table were obtained from the proposed circuit models withCc calculated from equation (3.54), Lo from the design formulas in [35] and Cs, Lsfrom the expressions in the Appendix of Chapter 2. As can be seen, a reasonableagreement between theory and experiment is obtained. It is remarkable that theCSRRs always resonate at frequencies slightly higher than those of the SRRs. Thiseffect is sharper for the higher dielectric constants.

FIGURE 3.34 Frequency response obtained in a rectangular waveguide loaded with SRRsand NB-SRRs, as well as their dual counterparts (C-SRR and C-NB-SRR). The method ofexcitation is sketched in the insets of the Figures. The element parameters are those ofTable 3.1. (Source: Reprinted with permission from [29]; copyright 2005, IEEE.)


3.8 SYNTHESIS OF METAMATERIAL TRANSMISSION LINES BYUSING CSRRs

In this section, it is demonstrated that CSRRs can be used for the design of meta-material transmission lines. From duality, it follows that whereas the dominantmode of excitation of SRRs is by applying an axial time-varying magnetic field,for CSRRs the main driving mechanism is electric coupling, and a significant com-ponent of the electric field parallel to the axis of the rings is required. Thus, by etchingthe CSRRs in the ground plane (in close proximity to the conductor strip), or in theconductor strip (provided there is space enough for this purpose), the requiredconditions for CSRR excitation are fulfilled in most common transmission lines(microstrip, CPW, and so on).

3.8.1 Negative Permittivity and Left-Handed Transmission Lines

The natural (although not exclusive) host transmission line for the implementation ofone-dimensional metamaterials using CSRRs is the microstrip configuration. Byetching the complementary rings in the ground plane, under the signal strip, a signifi-cant component of the electric line field results parallel to the rings’ axis, as desired.Figure 3.35 illustrates such structure, where 4 CSRRs are etched in the ground plane(relevant dimensions are indicated in the caption). The simulated and measured fre-quency responses for this device are also depicted in Figure 3.35. Due to the negativeeffective permittivity in the vicinity of CSRR’s resonance, the signal is inhibited in anarrow band.

In order to synthesize a left-handed medium, additional elements able to providethe required negative effective permeability must be introduced to the structure. It wasshown in the beginning of this chapter that the negative effective permeability can beachieved by etching series capacitive gaps in the host line. These gaps make thestructure behave as a magnetic plasma, with negative valued permeability up to afrequency that depends on the resonator formed by the gap capacitance and the

TABLE 3.1 Measured and Theoretical Values for the Frequency of Resonance

Conventional Complementary

f th0 (GHz) f exp0 (GHz) f th0 (GHz) f exp0 (GHz)

SRRa 7.17 7.40 7.49 8.00NB-SRRa 7.17 7.56 7.49 8.14DSRa 5.07 5.05 5.30 5.492-SRa 3.59 3.78 3.75 4.07SRRb 3.33 3.40 3.56 3.772-SRRb 6.66 6.77 7.12 7.41

The resonators are printed on a substrate with thickness h ¼ 0.49 mm and relative permittivity 1r ¼ 2.43.The parameters of rings, named in Figure 3.28, are (a) ro ¼ 1.7 mm, c ¼ d ¼ 0.2 mm; (b) ro ¼ 3.55 mm, c ¼d ¼ 0.3 mm.

3.8 SYNTHESIS OF METAMATERIAL TRANSMISSION LINES BY USING CSRRs 163

FIGURE 3.35 Bottom face (ground plane) of a microstrip structure loaded with CSRRs (a),simulated (b) and measured (c) frequency responses. CSRR dimensions are c ¼ d ¼ 0.3mm,and the radius of the inner ring is r ¼ 2.1mm. The period of the structure is 7mm. The substrateused is the commercially available Rogers RO3010 (1r ¼ 10.2, thickness h ¼ 1.27mm).(Source: Reprinted with permission from [31]; copyright 2004, IEEE.)


per-section inductance of the line. Hence, to design a left-handed metamaterial bycombining CSRRs and gaps, it is necessary to design the structure such that theplasma frequency extends beyond the resonance frequency of the CSRRs. Bytaking this into account, a region where negative effective permeability and permit-tivity coexist is expected. Figure 3.36 illustrates a left-handed microstrip line aswell as its measured frequency response. A passband with negative wave propagationarises. As occurs in SRR-based left-handed transmission lines, a sharp cutoff in thelower band edge and a soft upper transition band are visible. The reasons for thisbehavior are discussed in the following subsection, where the equivalent circuitmodels of these structures are inferred.

We have indicated that the usual host line for the design of CSRR-based 1D planarmetamaterials is the microstrip line. However, it does not mean that CPWs and otherplanar transmission lines should be ruled out. Indeed, by using CPWs, the lateraldimensions necessary to achieve a matched line are not univocally determined, andhence, the central strip can be made wide enough to provide the required space toetch the CSRRs on it. This way, the ground plane can be preserved from beingetched, which is of interest in certain applications.23 On the other hand, the

FIGURE 3.36 Layout of a left-handed microstrip structure based on CSRRs (a) andmeasured frequency response (b). The period of the device, the strip width and gap spacingare l ¼ 6mm, W ¼ 1.2mm and lg ¼ 0.2mm, respectively. The dimensions of CSRRs havebeen determined to obtain a resonant frequency of fo ¼ 3.5GHz, that is, c ¼ d ¼ 0.3mmand the internal radius of the inner ring is r ¼ 1.6mm (the parameters of the RogersRO3010 substrate have been considered, namely, dielectric constant 1r ¼ 10.2, thicknessh ¼ 1.27mm).


23For instance, if the substrate is sustained by a metallic holder.

etching of CSRRs in the signal strip of microstrip lines is possible in applicationswhere low-impedance transmission line sections are required (as will also beshown in the next chapter).

3.8.2 Equivalent Circuit Models for CSRR-Loaded Transmission Lines

Because CSRRs are mainly excited by the electric field induced by the line, this coup-ling can be modeled by series connecting the line capacitance to the CSRRs, whichare modeled as parallel LC tanks, as has been shown previously. According to this,the proposed lumped element equivalent circuit model for the CSRR-loaded trans-mission line is that depicted in Figure 3.37, where L and C are the per-section induc-tance and capacitance of the line, and Lc and Cc model the CSRR. For the structureloaded with CSRRs and series gaps (left-handed medium), the lumped elementequivalent circuit model is identical, but with the addition of a capacitance, Cg, toaccount for the series gaps (see Fig. 3.38).24 In fact, for these models to be valid,the CSRRs should be relatively close. Otherwise, a significant portion of the linewould lie outside the region occupied by the CSRRs and the approximation of thecoupling capacitance by the line capacitance would not be correct [36].25

For an infinitely periodic structure composed by cascading the T-circuit models ofFigures 3.37 and 3.38, the dispersion relation and Bloch impedance are given byexpressions (3.1) and (3.2), respectively. The dispersion relations for both the nega-tive permittivity and left-handed transmission lines inferred from the correspondinglumped element circuit models are depicted in reduced Brillouin diagrams inFigure 3.39 (these diagrams have been obtained by using the electrical parameters

FIGURE 3.37 Equivalent circuit model of a negative-permittivity line consisting of a trans-mission line loaded with CSRRs.

24It must be mentioned that with the series gaps etched above the position of the CSRRs, C accounts for theline capacitance and the fringing capacitance. As will be shown later, C may change substantially when thegaps are present.25A more accurate model for CSRR/gap loaded transmission lines, which includes line effects betweenadjacent CSRRs (substantially spaced CSRRs), as well as inter-resonator coupling (tiny spaced CSRRs),has been proposed in [36]. See also Problem 3.8.


corresponding to the microstrip structures of Figs 3.35 and 3.36). The features aresimilar to those of Figure 3.26, corresponding to negative permeability and left-handed media, respectively, implemented through SRRs. For comparison, thedispersion relation inferred from the simulated S-parameters of the unit cell isalso depicted in the diagrams. The slight discrepancies are mainly due tofabrication-related tolerances. These diagrams point out the left-handed nature ofthe line loaded with CSRRs and series gaps, as well as the stopband behaviorof the structure without the presence of the gaps, which is related to the negativeeffective permittivity associated with the CSRRs.

Let us now focus on the analysis of the transmission properties of CSRR left-handed transmission lines to the light of the equivalent circuit model. The structure

FIGURE 3.38 Equivalent circuit model of a left-handed line consisting of a transmissionline loaded with CSRRs and series gaps.

FIGURE 3.39 Theoretical (solid lines) and simulated (dashed lines) dispersion diagram forthe infinite CSRR–microstrip structures with unit cells identical to those shown in Figures 3.35and 3.36. The mismatch between the stop- and passbands is due to the different dimensions ofthe CSRRs in Figures 3.35 and 3.36. (Source: Reprinted with permission from [29]; copyright2005, IEEE.)


supports propagating waves in the frequency interval where f is a real number.Analysis of expression (3.1) indicates that this occurs in the region delimitedby the following frequencies (it has been assumed that the series impedance of theequivalent circuit model is dominated by Cg and, hence, L has been neglected inthe analysis)26:

fL ¼ 12p

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLc Cc þ

41Cg

þ 4C

0BB@

1CCA

vuuuuut

(3:56)

fH ¼ 1

2pffiffiffiffiffiffiffiffiffiffiLcCc

p , (3:57)

where the subindexes simply make reference to the lower and higher frequency of theinterval. At these frequencies, both the phase and Bloch impedance take extremevalues; that is, ZB !1 and f ¼ 0 at fH, whereas ZB ¼ 0V and f ¼ 2p at fL.Obviously, these results are valid under the assumption of negligible losses.

Inspection of the circuit of Figure 3.38 also reveals that there is a transmission zeroat a frequency given by

fz ¼1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLc(C þ Cc)

p : (3:58)

As long as C � 4Cg, fz and fL are very close, and the structure exhibits a very sharpcutoff in the lower edge of the left-handed allowed band, regardless of the number ofstages. As for the left-handed structure based on SRR, the number of transmissionpeaks (losses neglected) coincides with the number of stages. One of these trans-mission peaks arises as a consequence of impedance matching (that is, for that fre-quency that makes the Bloch impedance equal to ZB ¼ 50V), and the others aredue to phase matching, the exact location of them being given by those frequenciesfor which the phase variation satisfies the rule given in expression (3.52).

Following expressions (3.56) and (3.57), to design left-handed transmissionlines with relatively wide allowed bands, it is necessary that the gap capacitance aswell as the capacitance of the line is high, as compared to the equivalent capacitanceof the CSRRs. By using the following electrical parameters, that is, Cg ¼ 0.50pF,C ¼ 3.12pF, Lc ¼ 0.93nH, and Cc ¼ 2.81pF, the simulated electrical response ofthe equivalent T-circuit model (10-stage device) shown in Figure 3.38 can be inferred(Fig. 3.40). A left-handed microstrip structure with these electrical parameters hasbeen synthesized (Fig. 3.40). The geometry of the structure has been obtained

26This approximation is valid if the series resonance (i.e., the resonance frequency of the Cg and Lcombination) is substantially higher than the resonance frequency of the CSRRs (expression 3.57).


FIGURE 3.40 Fabricated 10-stage prototype device (a), insertion losses (b), and returnlosses (c). The period is l¼6.4mm, gap spacing g ¼ 0.2mm, and strip width W ¼ 1.69mm.The dimensions of the CSRRs are cin ¼ 0.472mm, cout ¼ 0.391mm, d ¼ 0.149mm, andrext ¼ 2.972mm. In (a) top and bottom views are shown. The device has been fabricated onthe Rogers RO3010 substrate (dielectric constant 1r ¼ 10.2, thickness h ¼ 1.27mm, tan d ¼

0.0023). (Source: Reprinted with permission from [37]; copyright 2006, John Wiley & Sons.)


according to the procedure described in [37]. The simulated and measured frequencyresponses of this structure are also shown in Figure 3.40. Obviously, the transmissionpeaks are obscured in the measurement due to conductor and dielectric losses.Nevertheless, these results point out the validity of the lumped element model andshow that left-handed structures with moderate left-handed bandwidth can beimplemented by using CSRRs.

3.8.3 Parameter Extraction

To further verify the validity of the lumped element circuit models of the CSRR-loaded lines, a method to extract the electrical parameters from either electromagneticsimulation or measurement has been proposed [38]. As will be shown at the end ofthis section, the agreement between measurement and electrical simulation of thecircuit models with extracted parameters is excellent. Indeed, the technique includesthe determination of the unloaded Q factor of the CSRRs. Hence, it is necessary toaccount for CSRR losses by adding a parallel resistance, R, to the models (Fig. 3.41).

In a first-order approximation, losses can be neglected and two characteristicfrequencies can be identified: the frequency that nulls the shunt impedance (trans-mission zero frequency, fz, given by expression 3.58) and the frequency that nullsthe shunt admittance (which obviously coincides with the intrinsic resonant fre-quency of the CSRR, fo, or fH according to expression 3.57). These frequenciescan be experimentally determined, or they can be obtained through full-wave

FIGURE 3.41 Topologies of the considered left-handed (a) and negative-permittivity (b)cells, and their equivalent circuit models including losses (c) and (d ), respectively. CSRRdimensions are (in reference to Fig. 3.28) c ¼ 0.3mm, d ¼ 0.19mm, rext ¼ 5.65mm, gap sep-aration and width are 0.30mm and 3.85mm, respectively, and the width of the access lines is1.15mm. Substrate characteristics correspond to the Rogers RO3010 substrate: dielectricconstant 1r ¼ 10.2, thickness h ¼ 1.27mm and tan d ¼ 0.0023. (Source: Reprinted with per-mission from [38]; copyright 2006, IEEE.)


electromagnetic simulation. At fz a notch in the transmission coefficient is expectedand this frequency can be measured accurately. To obtain fo, a representation of thetransmission coefficient on a Smith Chart is required. At this frequency the shunt pathto ground is opened, and the input impedance seen from the ports is solely formed bythe series elements of the structure and the resistance of the opposite port (50V).Therefore, fo is given by the intersection between the measured (or simulated) S11curve and the unit normalized resistance circle. From this result we can also obtainthe impedance of the series elements at that frequency. This directly gives thevalue of L for negative-permittivity lines. For left-handed lines, L can be indepen-dently estimated from a transmission line calculator, whereas Cg can be determinedby adjusting its value to fit the impedance inferred from simulation or experiment.Alternatively, to determine L and Cg in a left-handed line, a negative permittivityline can be fabricated, identical to the left-handed line except for the absence ofthe gap. By applying the technique to the structure without the series gap, L canbe univocally determined. It can then be corrected to account for the presence ofthe gap, and we can adjust Cg as mentioned before.

Expressions (3.57) and (3.58) are dependent on three parameters. Therefore, theelement values of the CSRR and the coupling capacitance cannot be directlyobtained. To this end, an additional condition is needed. This can be obtainedfrom the dispersion relation for the Bloch modes (3.1). At the angular frequency,vp/2, where the phase of the transmission coefficient (which is a measurablequantity) is f(S21) ¼ p/2, the phase constant is bl ¼ p/2. Following expression(3.1), we have

Zs(vp=2) ¼ �Zp(vp=2): (3:59)

Thus, from equations (3.57), (3.58), and (3.59), we can determine the three reactiveelement values that contribute to the shunt impedance. To determine R, this parameteris swept until electrical simulations of the circuit model and measured frequencyresponses agree. In practice, this parameter is determined with good accuracy fromthe transmission coefficient of the negative-permittivity structure, because the rejec-tion level is very sensitive to this parameter. From this, the unloaded Q factor is givenby Q ¼ RvoCc.

The method has been applied to the negative-permittivity and left-handed cellsdepicted in Figure 3.41 (substrate characteristics and dimensions are indicated inthe caption). The electrical parameters have been obtained by applying the methodto the measured frequency responses (Fig. 3.42), and they are represented inTable 3.2. By using these electrical parameters, the electrical simulation of the struc-tures can be inferred. The results are also depicted in Figure 3.42 for easy comparison.Very good agreement between measurements and electrical simulation has beenobtained. This confirms the validity of the circuit model of CSRR-loaded trans-mission lines. The slight discrepancy between measurements and electromagneticsimulation is attributed to fabrication-related tolerances. For more details on theparameter extraction technique see [38].


A similar parameter-extraction technique can be applied to SRR-loaded trans-mission lines, by considering the circuit models depicted in Figures 3.23b and3.24b. However, this would not directly provide the electrical parameters of theSRRs, but L0s and C0

s. To determine Ls and Cs the value of M is needed, and it is dif-ficult to accurately obtain such a parameter (in particular in the microstrip configur-ation). Nevertheless, parameter extraction for the models of Figures 3.23b and 3.24bcan be of interest in certain applications, and follows a similar procedure to thatdescribed above for CSRR-loaded lines.

3.8.4 Effects of Cell Geometry on Frequency Response

In this subsection, the effects of varying gap separation, substrate thickness and thewidth of the host transmission line are analyzed and interpreted in the framework ofthe proposed circuit model. Figure 3.43 presents three frequency responses corre-sponding to different gap separations. As gap decreases, the upper band edge isnot substantially altered. However, the lower band edge is clearly displacedtowards lower values, resulting in a wider allowed band. This effect can be explainedby expressions (3.56) and (3.57). The upper edge of the band only depends on the

FIGURE 3.42 Magnitude (a) and phase (b) of the transmission coefficient for the structuresshown in Figure 3.41. Measurement, electromagnetic simulation, and electrical simulation, aredepicted in bold, thin, and dashed lines, respectively. (Source: Reprinted with permission from[38]; copyright 2006, IEEE.)


parameters of the resonant tank that models the CSRRs. Therefore, we do not expect amodification of the upper fall-off, because gap width reduction does not affect theintrinsic resonant frequency of the CSRRs. However, the lower edge depends onCg and this capacitance increases as gap separation decreases. This explains theshift of the band onset towards lower frequencies when gap separation is decreased.In view of these results, the left-handed band of these structures can be widened bytailoring gap separation.

Bandwidth also depends on the height of the substrate. The results are depicted inFigure 3.44. The thinner the substrate is, the wider the bandwidth is. However, in thiscase, both band edges are shifted to expand the left-handed band. The upper bandedge is moved towards higher frequencies because, the thinner the dielectric is, the

TABLE 3.2 Extracted Element Parameters for the Structures Shown in Figure 3.41a

Cg (pF) L (nH) C (pF) Cc (pF) Lc (nH) R (kV)

LH cell 1.27 4.96 19.58 4.01 3.22 1.391, 0 cell – 5.08 4.43 4.06 2.98 1.39

aNotice that the value of C in the left-handed cell is much larger than in the negative permeability cell. Thegap should be modeled by the series capacitance and fringing capacitances in a p-model configuration.Through p-T transformation, the effects of the line and fringing capacitances can be grouped in a singlecapacitance, C, whose value can be substantially much larger than the addition of the individual com-ponents. This has been verified by the authors in other microstrip left-handed structures based onCSRRs and series gaps.

FIGURE 3.43 Simulated insertion losses for a 10-stage device with different values of gap dis-tance, g, obtained by means of agilent momentum. The dimensions are: period l ¼ 6.4mm,strip width W ¼ 1.69mm. The dimensions of the CSRRs are cin ¼ 0.472mm, cout ¼ 0.391mm,d ¼ 0.149mm and rext ¼ 2.972mm. The parameters of the Rogers RO3010 substrate have beenconsidered (1r ¼ 10.2; h ¼ 1.27 mm). (Source: Reprinted with permission from [37]; copyright2006, John Wiley & Sons.)


higher the resonant frequency of the CSRRs [29], and hence fH, becomes. On theother hand, as substrate height is reduced, the coupling capacitance C is augmented,this having the effect of reducing fL (in spite of the fact that Cg is slightly reduced).

Finally, the effects of varying line width are presented. This modifies both Cg andC, and it may slightly affect the resonance frequency of the CSRRs through the influ-ence of the upper metallic strip on Lc and Cc. We expect a variation of the lower bandedge towards lower frequencies when line width is widened (because both C and Cg

are expected to increase), and a small (although probably nonnegligible) shift in theupper transition band. This behavior has been verified by electromagnetic simulation(Fig. 3.45).

According to the results shown in this section, it is clear that 1D left-handed struc-tures with moderate bandwidths based on CSRRs can be designed. To enhance band-width, thin substrates are preferred, as this favors the possibility of moving away fLand fH. As long as bandwidth, out-of-band rejection (mainly dependent on thenumber of stages), and in-band losses can be controlled, these structures can beapplied to the synthesis of frequency-selective structures of compact dimensions.Also, these structures can be of interest for the design of artificial transmissionlines, where impedance and phase shift can be independently (within certainlimits) controlled over wide margins, without the need to manipulate line widthand length (as occurs in conventional transmission lines). This has many potentialapplications, as will be shown in the next chapter.

FIGURE 3.44 Simulated insertion losses for a 10-stage device with different values ofsubstrate height, h, obtained by means of agilent momentum. The dimensions are period l ¼6.4mm, gap spacing g ¼ 0.2mm, strip width W ¼ 1.69mm. The dimensions of the CSRRsare cin ¼ 0.472mm, cout ¼ 0.391mm, d ¼ 0.149mm, and rext ¼ 2.972mm. The parametersof the Rogers RO3010 substrate, except thickness, which is the varying parameter, havebeen considered (1r ¼ 10.2). (Source: Reprinted with permission from [37]; copyright 2006,John Wiley & Sons.)


3.9 COMPARISON BETWEEN THE CIRCUIT MODELS OFRESONANT-TYPE AND DUAL LEFT-HANDED LINES

At the beginning of Section 3.5, it was pointed out that resonant-type and dual27

transmission lines exhibit similar behavior. In particular, it was mentioned that, asin dual transmission lines, there is also a forward wave band at high frequencies inresonant-type structures. Indeed, in the preceding sections devoted to the synthesisand analysis of SRR- and CSRR-loaded transmission lines, the main interest wason demonstrating the occurrence of left-handedness, and on interpreting this interms of circuit analysis. However, inspection of the circuit model (unit cell) of ahost line loaded with series capacitors and shunt inductors (Fig. 3.3) and thosecircuits describing the unit cells of SRR loaded (Fig. 3.24) and CSRR loaded(Fig. 3.38) left-handed lines, indicates that similar characteristics may be expectedin all cases (obviously with certain singularities that will be highlighted).

Let us start by comparing the circuit of Figure 3.38 (CSRR-loaded left-handedline) to that of Figure 3.3 (dual transmission line). The single difference between

FIGURE 3.45 Simulated insertion losses for a 10-stage device with different values ofstrip width, W, obtained by means of agilent momentum. The dimensions are periodl¼6.4mm, gap spacing g ¼ 0.2mm. The dimensions of the CSRRs are cin ¼ 0.472mm,cout ¼ 0.391mm, d ¼ 0.149mm, and rext ¼ 2.972mm. The parameters of the RogersRO3010 substrate have been considered (1r ¼ 10.2; h ¼ 1.27 mm). (Source: Reprinted withpermission from [37]; copyright 2006, John Wiley & Sons.)

27Along this section, by dual line the authors mean a host line loaded with series capacitors and shuntinductors. A composite right-/left-handed behavior arises in such lines, due to the effect of the parasiticelements of the host line. These lines are usually called composite right-/left-handed (CRLH) lines (aswas pointed out previously). However, this CRLH behavior is not exclusive to this type of lines. It isalso present in resonant-type (i.e., loaded with SRRs or CSRRs) left-handed lines. For this reason weuse the term dual to designate C–L loaded lines.

3.9 COMPARISON BETWEEN THE CIRCUIT MODELS 175

these circuits is the presence of the coupling capacitance, C, in the circuit ofFigure 3.38. The direct consequence of this is the presence of a transmission zeroto the left-hand side of the backward wave band.28 In the limit of very high coupling(C! 1) the transmission zero shifts to the origin and both models tend to be iden-tical. In practice the capacitance C can be enhanced by decreasing the substrate height(in microstrip configuration). Thus, we may conclude by mentioning that CSRR-loaded left-handed lines implemented in very narrow microstrip substrates can bedescribed by a circuit model that is formally identical to that of the dual transmissionline in the region of interest.

Another key aspect to highlight is the composite right-/left-handed (CRLH) beha-vior of CSRR-loaded lines [39,40]. The origin of such behavior in dual transmissionlines was studied in detail in Section 3.2. According to the model of Figure 3.38, thisbehavior is also expected in CSRR-loaded left-handed lines, because above the res-onance frequency of the series resonator (formed by L and Cg), the series impedanceis positive (inductive), whereas the shunt admittance is dominated by Cc. Indeed,balanced CSRR structures can be achieved by merely forcing the series (vs ¼

(LCg)21/2) and shunt (vp ¼ (LcCc)

21/2) resonances to be identical.29 This possibilityis demonstrated in Figure 3.46, which depicts the dispersion diagram of a balancedCRLH CSRR-loaded transmission line cell (also included in the figure). As can beseen, the structure is almost balanced (that is, there is a quasi-continuous transitionbetween the left handed and right handed bands).30 The measured and simulated fre-quency responses of the cell, also depicted in Figure 3.46, exhibit a very broad trans-mission band, which is due to the absence of gap between the backward and forwardregions.31 According to these results, it is very clear that CRLH and balanced CRLHtransmission lines can be achieved by loading the host line with CSRRs. Also, it isclear that broadband metamaterial transmission lines can be achieved by means ofthe resonant-type approach. It is interesting to analyze the dependence of the charac-teristic impedance on frequency for the CRLH CSRR-loaded line. From expression(3.2) and the circuit of Figure 3.38 (including all the circuit parameters), the followingexpression is obtained (Problem 3.3):

ZB ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

Cc

1� v2s

v2

� �

1�v2p

v2

!� L2v2

41� v2

s

v2

� �2þ L

C1� v2

s

v2

� �vuuuuuut : (3:60)

28This transmission zero is of interest for the design of planar filters based on CSRR left-handed lines, aswill be discussed in Chapter 4.29The shunt resonance frequency is the frequency that nulls the shunt admittance and it coincides with theintrinsic resonance frequency of the CSRRs. It also coincides with the upper limit of the left-handed bandin unbalanced structures (expression 3.57).30Perfect balance is difficult to achieve in practice. Nevertheless, for the structure of Figure 3.46, a continu-ous transition between the left-handed and right bands is visible in the frequency response.31The measured and simulated (through EM software) frequency responses are wider than that predicted bythe circuit model. The reason for this is the inability of the model to properly describe the structure at highfrequencies, as is discussed in [39,40].


FIGURE 3.46 Balanced CRLH cell based on a microstrip line loaded with CSRRs (a), dis-persion diagram (b) and frequency response (c). The structure has been implemented in theRogers RO3010 substrate with dielectric constant 1r ¼ 10.2 and thickness h ¼ 1.27mm.Dimensions are line width W ¼ 0.8mm, external radius of the outer ring r ¼ 7.3mm, ringwidth c ¼ 0.4mm, and ring separation d ¼ 0.2mm; the interdigital capacitor, formed by 28fingers separated by 0.16mm, has been used to achieve the required capacitance value.(Source: Reprinted with permission from [39]; copyright 2007, IEEE.)


For the balanced case vs¼ vp ¼ vo, and the previous expression simplifies to

ZB ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

Cc� L2v2

41� v2

o

v2

� �2

þ L

C1� v2

o

v2

� �s: (3:61)

This expression is very similar to equation (3.22). The difference is the presenceof the last term in the square root of equation (3.61). As for the balanced dual trans-mission line, the characteristic impedance is null at the extremes of the allowed band,and it varies continuously in between. However, in resonant-type balanced linesloaded with CSRRs, the impedance is maximum above the transition frequency, fo(see Problem 3.3). Nevertheless, the impedance variation in the vicinity of the tran-sition frequency is smooth and this is interesting to preserve matching in a wide band.The variation of Zp, Zs and ZB with frequency for the transmission line model ofFigure 3.38 in the balanced case is depicted in Figure 3.47.

With regard to the lumped element model of SRR-loaded left-handed lines(Fig. 3.24b), this can also be considered an extension of the dual transmission linemodel of Figure 3.3. Both models are identical in the limiting case of L0s ! 1.However, in practice, it is not possible to achieve high values of L0s, because M islimited and Cs cannot be made very large (see expression 3.39). In other words,the transmission zero cannot be easily driven to the origin. This is obvious,because this transmission zero directly coincides with the resonance frequency ofthe SRRs, which is identical to the resonance frequency of the parallel tank

FIGURE 3.47 Representation of the series, Zs, shunt, Zp, and characteristic impedance, ZB,for a CRLH transmission line corresponding to the model of a balanced CSRR-based structure.Electrical parameters are L ¼ 45 nH, Cg ¼ 0.48 pF, C ¼ 10 pF, Cc ¼ 3 pF, Lc ¼ 3 pF. Thedepicted values of Zs and Zp are the reactances. The transition frequency has been set tofo ¼ 1.5GHz. (Source: Reprinted with permission from [39]; copyright 2007, IEEE.)


formed by L0s and C0s. It has already been pointed out in Section 3.6 the CRLH

behavior of transmission lines loaded with SRRs and shunt inductors (strips orvias). For SRR-loaded lines, it is also possible to balance these lines. To this end,the structure must be tailored in order to have identical shunt (expression 3.42) andseries (expression 3.43) resonance frequencies.

According to the previous comments, dual and resonant-type metamaterial trans-mission lines are conceptually very similar. The narrow band behavior usually exhib-ited by SRR- or CSRR-based left-handed lines is not something intrinsic to theapproach, but rather due to the difficulty of tailoring the geometry of the lines (reso-nators and other elements) to the required values to achieve wide left-handed bands.This narrow-band behavior also occurs (though less accentuated) in fully planar dualtransmission lines. To obtain wide left-handed bands, lumped elements may be used.Alternatively, to synthesize broad transmission bands, balanced lines can beimplemented. These balanced lines are also convenient to ensure impedance match-ing in a wide band, although, obviously, this band includes a backward and a forwardregion. The balance condition can also be forced in resonant-type CRLH transmissionlines. Hence, it is potentially possible to achieve broadband characteristics by usingSRR- and CSRR-loaded lines. In other words, broadband characteristics are notexclusive of the dual transmission line approach.

To end this chapter we would like to compare the lumped element model of SRR-loaded left-handed transmission lines (Fig. 3.24) with the transmission line model ofbulk left-handed structures consisting of a combination of SRRs and metallic posts(see Fig. 2.20). This model was described in [41], and it coincides with the modeldepicted in Figure 3.24b. In such a model, the posts are accounted for by means ofshunt inductors, whereas SRRs and their magnetic excitation by the incident fieldare appropriately described by means of series-connected parallel LC resonators. Itwas shown in [41] that through the transmission line model depicted inFigure 3.24b,32 the effective permeability and permittivity, as derived fromexpressions (3.15) and (3.16), coincide with the effective permittivity and per-meability of the SRR/wire medium, reported in [42]:

1eff ¼ 1� vp

v2(3:62)

meff ¼ 1� Fv2o

v2 � v2o � jvG

, (3:63)

where vp is the plasma frequency of the wire medium, vo is the resonance fre-quency of the SRRs, F is the fractional area occupied by the SRRs in the unitcell, and G is the dissipation factor, which depends on the conductor losses inthe SRRs. It is interesting to highlight that by modeling the SRR-loaded

32In the transmission line model of the SRR/wire medium considered in [41], losses have been included bymeans of a resistance, disposed in parallel with the resonant tank describing the SRR.


left-handed line by means of the circuit of Figure 3.24a, where the lumpedelements have a very clear physical interpretation, we finally obtain (through trans-formation 3.35–3.41) a circuit model (Fig. 3.24b) that is identical to that circuitneeded to obtain the effective electromagnetic parameters (1eff and meff) of theSRR/wire medium. This means that the bulk SRR/wire media and the planarSRR-loaded left-handed lines are equivalent structures, because they are describedby identical transmission line circuit models. Thus, the models of Figure 3.24 andthe transformation equations (3.35–3.41) between them bridge the gap between thebulk and transmission line approaches for the analysis of SRR-based left-handedmetamaterials.

PROBLEMS

3.1. Equivalence between plane wave propagation in homogeneous dielectricsand TEM wave propagation in transmission lines. Demonstrate the equival-ence between plane wave propagation in homogeneous isotropic dielectrics andTEM wave propagation in transmission lines given by expressions (3.15) and(3.16), and use it to obtain the effective permittivity and permeability of ideallossless conventional, dual, and CRLH transmission lines.

3.2. Phase constant of a CRLH line. Obtain the phase constant of a CRLH trans-mission line in the long wavelength limit (expression 3.29) by expandingexpression (3.21) in a Taylor series.

3.3. Analysis of the CSRR-based left-handed lines by including the effects of theline inductance. The CSRR-based left-handed transmission line is described bythe lumped element model represented in Figure 3.38. In this model, the seriesinductance of the line has been neglected. This can be done provided (a) theright-handed band is significantly separated from the left-handed band and (b)the interest is only focused on the left-handed transmission band. However, ina general case, it may be of interest to exploit the composite right-/left-handedbehavior of such lines. In this case the complete circuit model (with L included)must be considered. From this model, obtain the expressions for the character-istic impedance and phase constant. Simplify these expressions for thebalance condition and determine in this case where the characteristic impedancereaches its maximum.

3.4. Hybrid model of left-handed lines based on CSRRs. The following circuitmodel corresponds to the hybrid approach of a left-handed transmission line(basic cell), because it describes a transmission line loaded with both CSRRs(modeled by Cc and Lc) and shunt inductors (Lp). The series capacitances Cg

are responsible for the negative effective permeability of the line, as occurs inthe dual transmission line and resonant-type approach. For this model, obtainthe different transmission regions indicating if propagation is backward orforward.


3.5. Repeat the previous problem, excluding the capacitance Cc.

3.6. CRLH lines based on SRRs. For the circuit model given in Figure 3.24b, thedispersion relation has been analytically obtained (expression 3.45). Derive thisexpression and demonstrate that above the left-handed pass band, there exists aforward wave transmission band. Obtain the conditions that are necessary tosuppress the frequency gap between the left-handed and right-handed bands(balance condition).

3.7. Relationship between SRR and CSRR parameters. Demonstrate that forinfinitely thin perfect conducting screens, and in the absence of any dielectricsubstrate, the parameters of the circuit models for the SRRs and the CSRRs(Fig. 3.28) are related by Cc ¼ 4(1o=mo)Ls and Co ¼ 4(1o=mo)Lo.

3.8. CSRR-based left-handed lines with inter-resonator coupling. A generaliz-ation of the equivalent circuit model of a left-handed transmission lineimplemented by means of CSRRs is given as follows [36]:

This model takes into account not only the presence of the CSRRs coupled to the hostline, but also inter-resonator coupling (important in tiny spaced square-shaped or rec-tangular CSRRs), which is modeled by means of the mutual capacitances CM.Moreover, this model accounts for those situations where the distance between adja-cent resonators is significant (negligible coupling) and the capacitance of the host linecorresponding to the regions where CSRRs are not present, C, must be considered.Assuming that CSRRs are closely spaced (C negligible), but coupling between

PROBLEMS 181

adjacent resonators is small, demonstrate that the previous model can be transformedto the following:

where L0(v) and C0(v) are frequency-dependent inductance and capacitance, respect-ively, related to the second and first term of the following impedance:

Zeq ¼1

jvCg

1� Lr(CC þ 2CM þ Cr)v2½ � Lr(CC þ Cr)v2 � 1½ �LL2rC

2CCMv

6 þ 1� Lr(CC þ 2CM þ Cr)v2½ �Lr(CC þ Cr)v2 � 1½ � � L2rC

2CCMv

4=Cg

þ jvLvCg 1� Lr(CC þ 2CM þ Cr)v2½ � Lr(CC þ Cr)v2 � 1½ �LL2rC

2CCMCgv

7 þ vCg 1� Lr(CC þ 2CM þ Cr)v2½ �Lr(CC þ Cr)v2 � 1½ � � L2rC

2CCMv

5

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2. A. A. Oliner “A periodic-structure negative-refractive-index medium without resonantelements.” URSI Digest, IEEE-AP-S USNC/URSI National Radio Science Meeting, SanAntonio, TX, pp. 41, June 2002.

3. C. Caloz and T. Itoh “Application of the transmission line theory of left-handed (LH)materials to the realization of a microstrip LH transmission line.” Proc. IEEE-AP-SUSNC/URSI National Radio Science Meeting, vol. 2, San Antonio, TX, pp. 412–415,June 2002.

4. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in CommunicationElectronics. Wiley, New York, 1994.

5. J. A. Kong Electromagnetic Wave Theory. EMW Pub., 2000.


7. C. Caloz and T. Itoh “Novel microwave devices and structures based on the transmissionline approach of metamaterials.” IEEE-MTT Int’l Microwave Symp., vol. 1, Philadelphia,pp. 195–198, June 2003.


8. C. Caloz and T. Itoh Electromagnetic Metamaterials: Transmission Line Theory andMicrowave Applications. Wiley, Hoboken, NJ, 2006.

9. C. Caloz, H. Okabe, T. Iwai, and T. Itoh “Transmission line approach of left handedmaterials.” USNC/URSI National Radio Science Meeting, San Antonio, TX, vol. 1,pp. 39, June 2002.

10. A. Grbic and G. V. Eleftheriades “Experimental verification of backward wave radiationfrom a negative refractive index metamaterial.” J. Appl. Phys., vol. 92, pp. 5930–5935,November 2002.

11. G. V. Eleftheriades and K. G. Balmain Negative Refraction Metamaterials: FundamentalPrinciples and Applications. Wiley, Hoboken, NJ, 2005.

12. R. Shelby, D. R. Smith, and S. Schultz “Experimental verification of a negative index ofrefraction.” Science, vol. 292, pp. 77, 2001.

13. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer “Planar negative refractive index mediausing periodically L-C loaded transmission lines.” IEEE Trans. Microwave Theory Tech.,vol. 50, pp. 2702–2712, December 2002.

14. A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades “Experimental and theoretical verifica-tion of focusing in a large, periodically loaded transmission line negative refractive indexmetamaterial.” Opt. Express, vol. 11, pp. 696–708, April 2003.

15. A. Grbic and G. V. Eleftheriades “Overcoming the diffraction limit with a planar lefthanded transmission line lens.” Phys. Rev. Lett., vol. 92, paper 117403, March 2004.

16. A. Sanada, C. Caloz, and T. Itoh “Planar distributed structures with negative refractiveindex.” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 1252–1263, April 2004.

17. D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch “High-impedance electromagnetic surfaces with a forbidden frequency band.” IEEE Trans.Microwave Theory Tech., vol. 47, pp. 2059–2074, November 1999.

18. A. Grbic and G. V. Eleftheriades “An isotropic three-dimensional negative refractive indextransmission line metamaterial.” J. Appl. Phys., vol. 98, paper 043106, August 2005.

19. W. J. R. Hoefer, P. P. M. So, D. Thompson, and M. Tentzeris “Topology and design ofwide band 3D metamaterials made of periodically loaded transmission line arrays.”IEEE MTT Int. Microwave Symposium Digest, pp. 313–316, Long Beach, CA, June 2005.

20. F. Martın, F. Falcone, J. Bonache, R. Marques, and M. Sorolla “Split ring resonatorbased left handed coplanar waveguide.” Appl. Phys. Lett., vol. 83, pp. 4652–4654,December 2003.

21. I. Gil, J. Bonache, J. Garcıa-Garcıa, F. Falcone, and F. Martın “Metamaterials in microstriptechnology for filter applications.” Proc. APS-URSI, Washington, July 2005.

22. F. Falcone, F. Martin, J. Bonache, R. Marques, and M. Sorolla “Coplanar waveguide struc-tures loaded with split ring resonators.” Microwave Opt. Tech. Lett., vol. 40, pp. 3–6,January 2004.

23. R. Marques, F. Mesa, J. Martel, and F. Medina “Comparative analysis of edge and broad-side couple split ring resonators for metamaterial design. Theory and experiment.” IEEETrans. Antennas Propag., vol. 51, pp. 2572–2582, October 2003.

24. F. Falcone, F. Martın, J. Bonache, R. Marques, T. Lopetegi, and M. Sorolla “Left handedcoplanar waveguide band pass filters based on bi-layer split ring resonators.” IEEEMicrowave Wirel. Comp. Lett., vol. 14, pp. 10–12, January 2004.

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25. R. Marques, F. Medina, and R. Rafii-El-Idrissi “Role of bianisotropy in negative per-meability and left handed metamaterials.” Phys. Rev. B, vol. 65, paper 144440, April 2002.

26. J. D. Baena, R. Marques, and F. Medina “Artificial magnetic metamaterial design by usingspiral resonators.” Phys. Rev. B, vol. 69, paper 14402 (2004).

27. F. Falcone, F. Martın, J. Bonache, M. A. G. Laso, J. Garcıa-Garcıa, J. D. Baena,R. Marques, and M. Sorolla “Stop band and band pass characteristics in coplanar wave-guides coupled to spiral resonators.” IEEE Microwave Opt. Tech. Lett., vol. 42, pp.386–388, September 2004.

28. G. Zhang, F. Huang, and M. J. Lancaster “Superconducting spiral filters with quasi-ellipticcharacteristic for radio astronomy.” IEEE Trans. Microwave Theory Tech., vol. 53, pp.947–951, March 2005.

29. J. D. Baena, J. Bonache, F. Martın, R. Marques, F. Falcone, T. Lopetegi, M. A. G. Laso,J. Garcıa, I. Gil, and M. Sorolla “Equivalent circuit models for split ring resonators andcomplementary split rings resonators coupled to planar transmission lines.” IEEE Trans.Microwave Theory Tech., vol. 53, pp. 1451–1461, April 2005.

30. B. C. Wadell Transmission Line Design Handbook. Artech House, Norwood, MA, 1991.

31. F. Falcone, T. Lopetegi, J. D. Baena, R. Marques, F. Martın, and M. Sorolla “Effectivenegative-1 stop-band microstrip lines based on complementary split ring resonators.”IEEE Microwave Wirel. Comp. Lett., vol. 14, pp. 280–282, June 2004.

32. H. G. Booker “Slot aerials and their relation to complementary wire aerials (Babinet’sprinciple).” J. IEE, vol. 93, pt. III–A, no. 4, pp. 620–626, March–May 1946.

33. G. A. Deschamps “Impedance properties of complementary multiterminal planar struc-tures.” IRE Trans. Antennas Propag., vol. AP–7, pp. 371–378, December 1959.

34. W. J. Getsinger “Circuit duals on planar transmission media.” 1983 IEEE MTT-SInternational Microwave Symposium, Boston, May–June 1983, pp. 154–156.

35. I. Bahl and P. Bhartia Microwave Solid State Circuit Design. Wiley, New York, 1988.

36. I. Gil, J. Bonache, M. Gil, J. Garcıa-Garcıa, F. Martın, and R. Marques “Accurate circuitanalysis of resonant type left handed transmission lines with inter-resonator’s coupling.”J. Appl. Phys., vol. 100, paper 074908-1-10, October 2006.

37. M. Gil, J. Bonache, I. Gil, J. Garcıa-Garcıa, and F. Martın “On the transmission propertiesof left handed microstrip lines implemented by complementary split rings resonators.”Int. Numerical Modelling: Electronic Networks, Devices and Fields, vol. 19, pp.87–103, 2006.

38. J. Bonache, M. Gil, I. Gil, J. Garcia-Garcıa, and F. Martın “On the electrical characteristicsof complementary metamaterial resonators.” IEEE Microwave Wirel. Comp. Lett., vol. 16,pp. 543–545, October 2006.

39. M. Gil, J. Bonache, J. Selga, J. Garcıa-Garcıa, and F. Martın “Broadband resonant typemetamaterial transmission lines.” IEEE Microwave Wirel. Comp. Lett., vol. 17, pp.97–99, February 2007.

40. I. Gil, J. Bonache, M. Gil, J. Garcıa-Garcıa, and F. Martın “On the left handed andright handed transmission properties of microstrip lines loaded with complementarysplit rings resonators.” IEEE Microwave Opt. Tech. Lett., vol. 48, pp. 2508–2511,December 2006.


41. G. V. Eleftheriades, O. Siddiqui, and A. Iyer “Transmission line models for negativerefractive index media and associated implementations without excess resonators.” IEEEMicrowave Wirel. Comp. Lett., vol. 13, pp. 51–53, February 2003.

42. D. R. Smith and N. Kroll “Negative refractive index in left handed materials.” Phys. Rev.Lett., vol. 85, pp. 2933–2936, October 2000.

REFERENCES 185

CHAPTER FOUR

Microwave Applications ofMetamaterial Concepts

4.1 INTRODUCTION

The main aim of this chapter is to highlight the potential of metamaterials andmetamaterial-based structures for the synthesis and design of new microwavedevices and subsystems. Improved performance and novel functionalities, due tothe unique and controllable electromagnetic properties of these structures, as wellas device miniaturization, because of the small electrical size of their constitutiveelements, are the main areas where metamaterial concepts may produce a majorimpact. In the following subsections, several applications of metamaterials will beconsidered and exhaustively analyzed. Although this textbook is mostly orientedtowards the resonant-type approach of metamaterials, the authors wish to provide awide overview of metamaterial applications, including representative achievementsof other authors. Hence, several applications will be illustrated through implemen-tations based on the dual transmission line concept.

As metamaterials are frequency-selective structures by nature, their application tothe design of compact microwave filters and diplexers seems to be straightforward.Thus, the first subsection of this chapter is focused on the design of microwavefilters. Because such devices require resonant elements, the preferred (althoughnot exclusive) approach for the synthesis of filters and diplexers based on one-dimensional (1D) metamaterials is the resonant-type approach. It will be shown inSection 4.2 that SRRs and CSRRs are useful particles for the synthesis of narrow-band and wide-bandpass filters (also including bandpass filters for ultrawide-band—UWB—applications). A systematic methodology for the design of planar filterssubject to specifications will be presented. Finally, reconfigurable filters based onvaractor-loaded metamaterials will also be included in this section.


187

Probably the most outstanding property of metamaterial transmission lines is thecontrollability of the electrical characteristics. This includes the dispersion diagram1 aswell as the characteristic impedance of such artificial lines. Owing to this control, it ispossible to design components with superior performance compared to conventionalimplementations (such as enhanced bandwidth devices), components based on newfunctionalities (such as dual band components), or microwave devices with smalldimensions. In suchdevices, the conventional transmission lines and stubs are substitutedby metalines, resulting in metacircuits with smaller dimensions and/or improvedperformance. These metamaterial based components will be discussed and studied inSection 4.3. Itwill also be shown in this section that coupled lines basedonmetamaterialsare useful to improve the performance/dimensions of coupled-line couplers.

Radiated wave applications, due to the unusual dispersion diagram of left-handedor CRLH metamaterial transmission lines, will also be considered in this chapter. Itwill be shown that these structures may produce backfire (analogous to reversedCerenkov radiation in left-handed media [1]) to endfire radiation, including thepossibility of electronically scanning the radiation angle by tuning through varactordiodes. Implementations based on both the dual transmission line concept and theresonant-type approach will be provided.

4.2 FILTERS AND DIPLEXERS

It was shown in Chapter 3 that transmission lines simply loaded with SRRs or CSRRsare able to inhibit signal propagation in a narrow band in the vicinity of resonance[2,3]. This was interpreted as being due to the negative effective permeability orpermittivity of such 1D artificial transmission media. It was also demonstrated thatthis behavior can be switched to a passband by merely introducing additionalelements to the structures. Thus, in SRR-loaded lines, an allowed band withbackward-wave propagation can be generated by introducing shunt-connectedstrips or vias [4]. These elements provide the required negative effective permittivityup to a frequency (plasma frequency) that can be tailored through their dimensionsand separation. As long as this frequency is set beyond the resonance frequency ofthe SRRs, a frequency region where both electromagnetic parameters (permeabilityand permittivity) are negative arises, and hence a left-handed band. In artificial linesloaded with CSRRs, the additional elements must provide a negative-valued effectivepermeability (because the negative permittivity is obtained by means of the CSRRs),and this is achieved by etching series gaps to the line (as discussed in Chapter 3).The result is also a narrow band with negative wave propagation [5].

1In general, the dispersion characteristics in conventional transmission lines depend on both the substrateand line dimensions (width). However, it is not possible to tailor these characteristics as desired to achievecertain requirements. In metamaterial transmission lines, the dispersion diagram is determined by thecharacteristics of the host line and loading elements. This major number of line parameters makes it poss-ible to engineer such artificial lines in order to obtain the required characteristics (within certain limits). Forthis reason, the design of microwave components based on these artificial transmission lines can be calleddispersion engineering.

188 MICROWAVE APPLICATIONS OF METAMATERIAL CONCEPTS

In spite of their small dimensions, these structures have serious drawbacks for theirpractical applications as stopband or bandpass filters. The single negative SRR- orCSRR-loaded lines exhibit highly selective stopbands with a rejection level depend-ing on the number of stages [2]. However, it is difficult to control the width of theforbidden band. This depends on the distance between adjacent resonators, becauseinterresonator coupling enhances this gap. However, in practice, this coupling isvery limited, and no substantial control on device gap width is achieved by thismeans. As bandpass structures, the double-negative artificial lines considered inChapter 3 also exhibit some limitations. Namely, the typical frequency responsesmeasured on these left-handed lines are very selective at the lower edge of theband, due to the presence of a transmission zero, but the transition band is smoothat the upper band edge [4,5]. The frequency response is thus asymmetric, and thefirst spurious band may lie close to the passband of interest. On the other hand, band-width cannot be easily controlled and it is difficult to achieve wide bands. Dependingon the applications, the left-handed transmission lines studied in Chapter 3 maysuffice. However, in other applications, bandwidths beyond those that may be achiev-able by means of the cited structures, or frequency responses with wide stopbands, tomention some typical needs, may be required. The purpose of this section is to studyhow the structures presented in Chapter 3 can be modified in order to obtain compactfilters based on SRRs or CSRRs with improved performance. As will be shown, it ispossible to design both stopband and bandpass filters with small dimensions and elec-trical characteristics comparable to or superior than those that may be obtained bymeans of conventional distributed approaches. In the next subsection, the implemen-tation of stopband filters with controllable gap width is considered. After that, theimplementation of bandpass filters will be analyzed. The preferred strategies to gen-erate both narrow and wide bands will be discussed, including a design methodologybased on the generalized bandpass filter network. Also included in this study will bethe implementation of microwave diplexers based on CSRRs. The section devoted tofilters will end with a subsection focused on reconfigurable metamaterial transmissionlines and their application to tunable filters.

4.2.1 Stopband Filters

Stopband filters are typically of interest in microwave engineering for the suppressionof undesired responses or for the elimination of interfering signals. As has been indi-cated previously, single negative transmission lines based on SRRs or CSRRs arerejection band structures that produce reasonable suppression levels with smalldimensions. This is so because the elemental cell is very effective in inhibitingsignal propagation and is electrically small. However, for certain applications, the for-bidden bands required are wider than those achievable by cascading several SRR orCSRR stages. To overcome this limitation, we may take advantage of the high reflec-tivity of the particles considered, and cascade several resonators tuned at differentfrequencies over the frequency interval of interest. To this end, we may simplyscale up and down the dimensions of the SRRs (or CSRRs), or simply increase ordecrease their radius. Following this idea, it is possible to substantially enhance the

4.2 FILTERS AND DIPLEXERS 189

width of the forbidden band. This has been driven to practice in both microstrip andcoplanar waveguide technologies by using SRRs and CSRRs [6–8]. Figure 4.1 showsa multiple-tuned SRR-loaded CPW structure with five ring pairs. Tuning has beenimplemented in this case by increasing the inner radius of the SRRs in 0.05 mmstep increments and leaving c and d unaltered (as compared to the SRR geometryof Fig. 3.15). The simulated and measured frequency responses, depicted inFigure 4.1, show that the rejected band broadens towards lower frequencies. Thisis expected, because an increment of r has the effect of decreasing the resonant fre-quency of the SRRs. The pronounced cutoff at either side of the gap and the absenceof significant insertion losses in the pass band is noteworthy. This is important inusing SRRs for the elimination of undesired frequencies in CPW circuits.

The application of the multituning concept to SRR-loaded microstrip lines is alsopossible. In this case, square-shaped SRRs are preferred in order to enhance magneticcoupling between line and rings [7]. Figure 4.2 shows a possible implementation ofsuch a multituned structure. The host line is designed to exhibit a 50V impedance,and the SRR dimensions, indicated in the caption, have been varied by scaling upand down the geometry. The result is a wide stopband of approximately 1GHz

FIGURE 4.1 Layout (a), simulated (thin line), and measured (bold line) frequencyresponses (b) of the multituned SRR–CPW stopband filter. The length of the activeregion is 4 cm. The dimensions of the smallest SRRs are c ¼ d ¼ 0.2 mm, internal radiusr ¼ 1.3 mm. For the other SRRs, r has been incremented in 0.05 mm steps, and c and dhave been left unaltered. (Source: Reprinted with permission from [6]; copyright 2003, IEEE.)


centered at 4.5 GHz, as can be seen in the simulated and measured frequencyresponses, shown in Figure 4.2c and d. For comparison purposes, also included inthis figure is an electromagnetic bandgap (EBG)2 transmission line, which hasbeen designed to produce comparable rejection and gap width. The required

FIGURE 4.2 Multi-tuned SRR-based microstrip stopband filter compared to an EBGimplemented by varying the strip width of a microstrip line (a). Zoom view of the SRR-based structure (b), simulated frequency responses (c), and measured frequency responses(d ). The separation between concentric rings and their width are c ¼ d ¼ 0.3 mm for allSRRs. The larger SRRs have an external edge (in the direction orthogonal to the line) oflo ¼ 3.55 mm, and the smaller rings, lo ¼ 2.7 mm (tuning has been applied to this parameter).For all SRRs, the external edge in the direction of the line is ll ¼ 4.1 mm. Finally, the strip linewidth is W ¼ 1.17 mm and the distance to SRRs (outer edges) is 0.3 mm. The total length ofthe device is 39.3 mm, which corresponds to approximately 1.6l (l being the signal wave-length at the central frequency of the forbidden band). The structures have been fabricatedon the Rogers RO3010 substrate (1r ¼ 10.2, h ¼ 1.27 mm). (Source: Reprinted with per-mission from [7]; copyright 2005, John Wiley & Sons.)

2EBGs are periodic structures able to inhibit signal propagation at certain frequencies and/or directions[9,10]. They can be three-, two- or one-dimensional structures, and one particular case for the latter cat-egory corresponds to EBG transmission lines, where typically the ground plane [11] or the conductorstrip [12] are periodically microstructured to achieve the required properties (for instance by drillingholes in the ground plane [13], or by periodically modulating line width and hence line impedance[12,14–17]).


number of stages in the EBG structure (with linewidth modulation) is much higher.Because the period of the EBG structure scales with frequency (namely, it must be setto half the signal wavelength at the central frequency of the stopband), the resultingdimensions are huge. Therefore, compared to EBG transmission lines, the approachintroduced in this chapter, and based on SRRs, seems to be more effective not only interms of miniaturization but also to achieve superior performance [7].

As elements able to generate a negative effective permittivity, CSRRs are alsoexpected to produce controllable stopbands when etched in planar transmissionlines and tuned at different frequencies [8]. Examples of such structures implementedin microstrip and CPW technology are depicted in Figures 4.3 and 4.4, respectively.For the CSRR-loaded microstrip line (Fig. 4.3), rectangular-shaped CSRRs areetched in the ground plane under the conductor strip, where a significant componentof the electric field in the axial direction of the rings is present. This is necessary toachieve high electric coupling between line and CSRRs, which is in turn of interest toobtain high levels of rejection in the forbidden band. As in the previous illustrativeexamples, the host line must be matched to the ports to minimize insertion lossesin the allowed band. The relevant dimensions of the structure are indicated in thecaption. The measured frequency response of this structure, which is also depictedin Figure 4.3, reveals that rejection above 20 dB can be achieved within a 25% frac-tional band. For the CPW structure of Figure 4.4, similar levels of rejection can beachieved. In this case, however, the relative gap width is narrower, because asmaller number of CSRRs has been used. The reason is that CSRRs have been

FIGURE 4.3 50-V microstrip line with square-shaped CSRR etched in the ground plane (a)and measured frequency response (b). c ¼ d ¼ 0.4 mm and the external edges of the CSRRshave been tuned in the vicinity of 9.4 mm � 5.8 mm. The structure has been fabricated onthe Rogers RO3010 substrate (1r ¼ 10.2, h ¼ 1.27 mm). (Source: Reprinted with permissionfrom [8]; copyright 2005, IEEE.)


etched in the central strip. This represents the optimum procedure to save layout area,because the ground planes are maintained unaltered. This idea can find practicalapplication, as will be explained later.

To summarize, it has been shown that it is possible to implement quasiperiodicartificial transmission lines, loaded with SRRs or CSRRs, where, due to theircapability to inhibit signal propagation within a limited frequency interval, rejectionbands with controllable gap width can be generated. The most relevant characteristicof these structures is their size, which can be made small on account of the subwave-length nature of the constitutive elements (SRRs or CSRRs). This size can beoptimized by using other topologies discussed in Chapter 2, such as the BC-SRR,the SR, and so on. In the next subsection, the application of these concepts toimprove the frequency response of conventional distributed filters (through thesuppression of undesired spurious bands) is explored.

4.2.2 Planar Filters with Improved Stopband

The presence of spurious bands is an important limitation of microwave filtersimplemented by means of distributed elements [18]. These undesired frequencybands can seriously degrade filter performance and may be critical in certain appli-cations that require huge rejection bandwidths. Unfortunately, for most filterimplementations the first spurious band is relatively close to the frequency regionof interest. For example, in coupled line bandpass filters, the first spurious bandappears at the first harmonic of the central frequency (2fo), and it is a consequence

FIGURE 4.4 CPW stopband structure with CSRRs etched in the central strip (a), andmeasured frequency response (b). Ring dimensions are c ¼ d ¼ 0.4 mm and external edgesare tuned in the vicinity of 6.6 mm � 3.6 mm. The structure has been fabricated on theRogers RO3010 substrate (1r ¼ 10.2, h ¼ 1.27 mm). (Source: Reprinted with permissionfrom [8]; copyright 2005, IEEE.)


of the different phase velocities of the even and odd modes supported by the coupledlines. In capacitively (gap) or inductively coupled resonator bandpass filters, an unde-sired band is also inherently present at 2fo due to the resonance condition at this fre-quency. Finally, the stopband in stepped-impedance lowpass filters is limited by thepresence of the first spurious band, which is typically too close to the cutofffrequency. The rejection of these undesired frequency bands has been a subject ofinterest for filter designers for years. Traditional techniques include the use of half-wavelength short-circuit stubs, chip capacitors or cascaded stopband filters.However, these techniques are either narrow band, increase device area, or introducesignificant insertion losses. In coupled-line filters, several approaches based on modi-fied structures, aimed at obtaining equal modal phase velocities, have been proposedas a means to improve out-of-band filter performance [19–22]. These approaches arevery effective, but they are also very specific (that is, for application in parallelcoupled-line filters).

Based on the concept of EBGs [9,10], numerous works have been devoted to thesuppression of harmonics in a wide variety of microwave circuits, including passives[12,17,23] and actives [24]. Apart from this versatility, EBGs can be integrated withinthe device active region, thus avoiding the need to cascade additional stages[12,17,23]. This is an important property for optimizing final layout area.Although effective, frequency selectivity in EBG structures is based on their period-icity (Bragg effect), and several stages are required (typically 6 or 7) to obtain signifi-cant rejection levels. As the EBG period scales with wavelength (Bragg condition),the required dimensions of the structure might be too big for certain applications(in particular at moderate or low frequencies). Moreover, EBGs do not provide aneasy way to control gap width [25]. Nevertheless, EBG structures have been success-fully applied to the elimination of multiple spurious bands in microstrip bandpassfilters, with measured rejection levels above 30dB up to 5fo [23].

An alternative to the previously cited approaches for the elimination of spuriousfrequency bands in distributed filters is the use of SRRs or CSRRs. Their small elec-trical size, rejection efficiency, and versatility (stopband filters based on these par-ticles have been demonstrated in both microstrip and CPW technology [6,8]) makethese elements very attractive to achieve effective spurious suppression withminimum area increase. Such a possibility is explored in this subsection. The mainadvantage of the approach, as with EBGs, is the fact that SRRs and CSRRs can beetched within the active region of the devices. Namely, it is not necessary tocascade the SRRs or the CSRRs at the input or output access lines. This representsan optimum design in terms of area saving. To illustrate the efficiency of SRRsand CSRRs for the elimination of spurious bands and their versatility, several illustra-tive examples are provided. The first one is a parallel coupled-line bandpass filterimplemented in microstrip technology [8]. The layout of the device, a third-orderButterworth bandpass filter with a central frequency of fo ¼ 2.4 GHz and 10%fractional bandwidth, is depicted in Figure 4.5. Rectangular SRRs are etchedadjacent to the coupled lines in order to reject the first and second spurious bands.The multiple tuning procedure explained in the previous subsection has been used,although final SRR geometries have been determined by means of an optimization


algorithm (integrated within the Agilent momentum software). The smaller rings(etched in the central stages) are responsible for the rejection of the second spuriousband, whereas the first undesired band is rejected by the action of the larger SRRs,which are allocated in the first and fourth filter stages. The simulated and measuredfrequency responses of this device are represented in Figure 4.5b and c. In compari-son to a conventional device, the first spurious band is rejected with attenuation levels

FIGURE 4.5 (a) SRR coupled-line microstrip bandpass filter. (b) Simulated frequencyresponse compared to that obtained in the device without rings. (c) Measured frequencyresponses. The device has been fabricated on the Rogers RO3010 substrate (thicknessh ¼ 1.27 mm, dielectric constant 1r ¼ 10.2) by means of a standard photo/mask etching tech-nique. (Source: Reprinted with permission from [8]; copyright 2005, IEEE.)


near 40dBs, and insertion losses in the second spurious band are clearly below220dBs. This efficiency can be attributed to the significant number of SRR pairs dis-tributed along the device, which is possible by virtue of their small electrical size. It isalso worth mentioning that the use of square or rectangular SRRs enhances magneticcoupling between line and rings (as has been explained), and this allows for furtherrejection levels, as compared to circular SRRs [26,27].

In Figure 4.6, a coupled-line bandpass filter fabricated in CPW technology isdepicted [8]. In this device (a third-order Butterworth, 10% fractional bandwidth,

FIGURE 4.6 (a) CSRR coupled-line bandpass filter in CPW technology. (b) Simulatedfrequency response compared to that obtained in the device without rings. (c) Measured fre-quency responses. The device has been fabricated on the Rogers RO3010 substrate (thicknessh ¼ 1.27 mm, dielectric constant 1r ¼ 10.2). (Source: Reprinted with permission from [8];copyright 2005, IEEE.)


central frequency fo ¼ 1.8 GHz), square-shaped CSRRs are etched in the coupledlines with the aim of eliminating the first and second harmonic bands. Accordingto the frequency response, also shown in Figure 4.6, suppression levels above20dBs up to 5GHz are achieved, and the passband is scarcely affected. Thispoints out the efficiency of the technique, this time with CSRRs etched in theconductor strips. Compared with CPW coupled-line filters with SRRs etched inthe back substrate side (not shown), the structure of Figure 4.6 is more effectivefor spurious suppression. In addition, the back substrate side is left unaltered,except by the presence of the via bridges that are required in these types of structuresto avoid the generation of parasitic modes [28]. In fact, this structure (Fig. 4.6) isvery interesting, because the CSRRs are etched in the strips of the coupledlines; that is no lateral microstructuring is added to the device. Therefore this is avery interesting solution for the elimination of undesired bands in terms ofarea saving.

Following this idea, CSRRs may also be etched in the active region of otherplanar filters, such as stepped-impedance lowpass filters, in order to eliminate thefirst harmonic band [29]. The topology of these filters is very suitable for thispurpose because it presents low-impedance transmission line sections, which arewide and may provide the necessary space for CSRR etching. To illustrate this,the layout of a stepped-impedance lowpass filter with spurious suppression isdepicted in Figure 4.7. Circular CSRRs have been etched in the low-impedancesections, with the result of significant attenuation of the first harmonic band.Again, no area is added to the device because the resonators are etched in the metallicstrips of the filter.

In the reported examples, the passband of interest is not substantially affected bythe presence of the SRRs or CSRRs. This is an important aspect to highlight and it isrelated to the relatively high Q factor of the resonators. In other words, the signalignores the presence of the rings, unless its frequency is very close to the nominalresonant frequency of the SRRs (or CSRRs). These are tuned in the vicinity of thefirst (or second) spurious band and hence the frequency response in the operatingband is not substantially altered.3 Miniaturization (due to subwavelength operationof SRRs and CSRRs), high levels of rejection, and the possibility of controllinggap width by fine tuning, are key aspects that make this technique very promisingfor the elimination of undesired bands in microwave filters. Moreover, the techniquecan be applied to a wide variety of structures (including CPW and microstriptransmission lines) and filter types.

3In fact the presence of SRRs or CSRRs in the active region of the filters may affect somewhat the electricalcharacteristics of the structures. Specifically, it has been found that the presence of SRRs and CSRRs in theparallel coupled-line filters of Figures 4.5 and 4.6, respectively, slightly modifies the phase characteristicsof the coupled-line sections (as compared to the structures without SRRs or CSRRs). Therefore, it isnecessary to recalculate dimensions to preserve central filter frequency and bandwidth (the procedure isexplained in [8]). However, rather than, being a disadvantage, this is useful to reduce final device dimen-sions (5% and 10% for the filters of Figures 4.5 and 4.6, respectively).


4.2.3 Narrow Bandpass Filter and Diplexer Design

It has been indicated previously that left-handed transmission lines consisting of acombination of SRRs with shunt-connected strips (or vias), or CSRRs with seriesgaps, exhibit an abrupt transition band at the lower edge, but poor frequency selectiv-ity at the upper edge of the band. A transmission zero present below and close to theleft-handed allowed band explains the sudden change in the transmission coefficientprior to this band. However, the upper band limit is determined by that frequencywhere the series impedance switches from capacitive to inductive behavior (forSRR-based left-handed lines) or the shunt impedance switches from inductive tocapacitive behavior (for CSRRs lines).4 This change is gradual rather than abruptand this is the reason for the soft transition that is typically measured above the

FIGURE 4.7 (a) Layout of the CSRR-based stepped-impedance lowpass filter drawn toscale. Total device length, including 50-V access lines, is 94 mm; (b) Simulated (dashedlines) and measured (solid lines) frequency response for the CSRR stepped-impedance low-pass filter. The device has been fabricated on the Rogers RO3010 substrate (thickness h ¼1.27 mm, dielectric constant 1r ¼ 10.2). (Source: Reprinted with permission from [29].)

4Indeed, the left-handed transmission band can also be truncated at the upper edge by a change in the signof the shunt and series impedance for SRR and CSRR loaded lines, respectively. However, unbalancedresonant-type left-handed lines are usually designed with the resonance frequency of the branch not depen-dent on the SRRs or CSRRs well beyond SRRs’ or CSRRs’ resonance.


passband. One possibility to improve frequency selectivity at the upper edge of theband is to alternate left-handed transmission line sections with artificial right-handed sections. The latter, which consist of SRRs combined with gaps, orCSRRs combined with shunt-connected strips, exhibit a transmission zero abovethe passband. Therefore, by overlapping the passbands of both stages, it is potentiallypossible to achieve roughly symmetric and highly frequency-selective devices. Let usexplore this possibility.

4.2.3.1 Bandpass Filters Based on Alternate Right-/Left-Handed (ARLH)Sections Implemented by Means of SRRs A representative and illustrative struc-ture of the alternate right-/left-handed (ARLH) concept5 implemented by means ofSRRs is depicted in Figure 4.8 [30]. It consists of a CPW transmission line loadedwith three SRR pair sections. The central stage is a right-handed SRR/gap section,and it is sandwiched between two SRR/strip (i.e., left-handed) stages. The outerstages are described by the p lumped-element circuits of Figure 3.24, and thecentral stage is modeled by the circuits shown in Figure 4.9 (Cg is the capacitanceof the series gap). An analysis of this latter circuit indicates that signal propagationis allowed for those frequencies providing a real value of b, according to thefollowing expression:

cos(bl) ¼ 1þ C

2Cg� LCv2

2þ Cv

4L0s=C

0s

vL0s �1

vC0s

: (4:1)

Equation (4.1) corresponds to the dispersion relation of an infinitely long periodicstructure composed of a cascade of elemental cells. Although the number of cells isinfact limited (a single SRR/gap stage sandwiched between two SRR/strip cells isused in the structure of Fig. 4.8), equation (4.1) provides valuable information,because it confirms that the SRR/gap combination supports propagating modes in anarrowband below SRR resonance. Moreover, inspection of equation (4.1) revealsthat the series impedance rapidly changes from a highly inductive to a highly capacitivebehavior at resonance, with the result of a sharp cutoff at that frequency. This has beenverified by full-wave electromagnetic simulation of the central SRR/gap stage alone(the commercial software Agilent momentum has been used). The result is visible in

5Notice that the ARLH concept is different than the composite right-/left-handed (CRLH) concept intro-duced in Chapter 3. Namely, the CRLH concept expresses the composite (left- and right-handed) nature ofboth dual and resonant-type metamaterial transmission lines. The expression ARLH is used to designate thecombination of left-handed and right-handed sections in the same structure. Although the left-handed sec-tions exhibit a CRLH behavior, the right-handed band of such sections is not used. In ARLH structures, thealternance of right- and left-handed sections is not actually necessary. The relevant advantages of suchstructures are due to the combination of left- and right-handed sections. However the term combinedright-/left-handed has been avoided, because its acronym coincides with that of the composite right-/left-handed concept. Moreover, in the illustrative examples provided, the right-and left-handed sectionsdo infact alternate.


FIGURE 4.9 Lumped-element equivalent circuit for the gap-SRR stage (a) and simplifiedmodel with the series branch replaced by its equivalent impedance (b). Cg is the capacitanceof the series gap.

FIGURE 4.8 Layout of the strip/gap SRR-CPW structure. For the smaller SRRs the innerradius has been set to r ¼ 1.39 mm, and for the rings located at the edges of the structure,r ¼ 1.52 mm. In all cases, c ¼ d ¼ 0.2 mm. The strip width is ww ¼ 2.16 mm and the gaplength is lg ¼ 1.6 mm. With these dimensions, the shunt-connected inductance and series capa-citances take the values Lp ¼ 115.76 pH and Cg ¼ 96.84 fF, respectively. The lateral dimen-sions of the host CPW are W ¼ 5.4 mm, G ¼ 0.16 mm. The central strip is wide in order toavoid SRR-pair overlapping. The length of the active region of the device (i.e., excludingaccess lines and connectors) is 14.4 mm. The Arlon 250-LX-0193-43-11 substrate was used,that is, dielectric constant 1r ¼ 2.43, thickness h ¼ 0.49 mm. (Source: Reprinted with per-mission from [30]; copyright 2004, IEEE.)


Figure 4.10, where, for comparison purposes, the frequency response of an individualSRR/strip stage is also depicted. In both cases, the parameters of the Arlon 250-LX-0193-43-11 substrate were considered; that is, dielectric constant 1r ¼ 2.43, thicknessh ¼ 0.49 mm. Because in the allowed band the series impedance is inductive and theshunt impedance is capacitive, it follows that signal propagation is forward andhence the SRR/gap combination is a right-handed cell.

The different locations of the passband for the left-handed and right-handed stagesrelative to the resonant frequency of the rings explains the smaller dimensions of thecentral SRRs. The dimensions of the SRRs and other relevant dimensions of the filterare given in the caption of Figure 4.8. With these SRR geometries, the resonancefrequencies for the central and external SRRs (according to the model reportedin Chapter 2 and also given in [31]) are fo ¼ 7.22 GHz and fo ¼ 6.64 GHz,

FIGURE 4.10 Simulated frequency responses for individual SRRs/gap (a) and SRRs/strip(b) cells.


respectively.6 Hence, the passband of the structure is expected to lie within this fre-quency interval. The strip and gap geometries have been determined to obtain shuntinductances and a series capacitance able to obscure the effects of line capacitanceand line inductance in the region of interest (i.e., within the passband) for theouter and central stages, respectively. Filter size is approximately three timessmaller than the wavelength at the central frequency of the filter.

The simulated and measured frequency responses of the designed structure aredepicted in Figure 4.11. There is a reasonable agreement between simulation andexperiment. The slight discrepancies in the region of interest are attributed to fabrica-tion-related tolerances.7 In agreement with the analysis based on the equivalent circuitmodel, a passband is obtained between the resonant frequencies of the central andouter rings. The sharp transition band obtained at either edge of the allowed bandis noticeable, with more than 30 dB fall in 0.5 GHz and peak rejection levels of 60dBs and near 40 dBs at 6 GHz and 7.7 GHz, respectively.

Another illustrative example of the ARLH transmission line concept is an S-bandfilter (central frequency fo ¼ 2.4 GHz), where the number of cells has been reducedto two [32]. Namely, a single SRR/strip cell has been cascaded with an SRR/gapcombination (Fig. 4.12). The length of the active device region is 22.5 mm, that is,five times smaller than the signal wavelength at 2.4 GHz. The simulated and measuredfrequency responses of this device are depicted in Figure 4.13. In spite of the reducednumber of stages (only two), frequency selectivity is good, with measured rejectionlevels better than 50 dBs at the left of the pass band and above 30 dB up to approxi-mately 4 GHz. The upper transition band edge is very sharp with 60 dB fall in 0.4GHz, and an average slope of 125 dB/GHz has been measured at the lower band edge.

The potentiality of ARLH transmission lines in terms of filter miniaturization ispointed out by comparing the designed S-band filter with a conventional microstripcoupled-line bandpass filter with similar performance (an identical substrate isconsidered to validate the comparison). The layout of this filter is also depicted inFigure 4.12. The simulated frequency response of the coupled-line bandpass filteris depicted in Figure 4.13a (this device has not been fabricated). To obtain similarslopes in the transition bands, it has been necessary to design a three-stageconventional filter (Chebyshev response) with a length (excluding the access lines)of 63.6 mm, that is, roughly three times longer than the active region of the ARLHSRR-based prototype.8 The authors would like to highlight the absolute coincidenceof the simulated frequency responses in the pass band for both implementations(Fig. 4.13a). This indicates that in-band losses in the ARLH SRR filter are related

6Due to the presence of the CPW structure at the other side of the substrate, these resonance frequencies donot exactly coincide with those inferred from full-wave electromagnetic simulation (Fig. 4.10).7It has been demonstrated by simulation that variations of ring dimensions of less than 1% around thedesign values are enough to degrade in-band losses. This explains the measured in-band losses (4 dB)not being as good as those obtained by electromagnetic simulation, and also the slight frequency shiftbetween simulation and measurement.8The performance of the ARLH SRR-based filter in terms of frequency selectivity is superior. Therefore, atleast a four-stage coupled-line filter is necessary to improve the frequency response obtained in the SRR-based filter.


to the finite Q factor of the resonators, rather than to an improper design (although ithas been observed that in-band losses are very dependent on variations of ring dimen-sions around the design values). These results clearly suggest the possibilities of theARLH concept in reducing filter dimensions in planar technology.

4.2.3.2 Bandpass Filters and Diplexers Based on Alternate Right-/Left-Handed (ARLH) Sections Implemented by Means of CSRRs ARLH trans-mission lines properly designed to act as narrow bandpass filters can also beimplemented by means of CSRRs. As expected, the left-handed stages consist of

FIGURE 4.11 Simulated (a) and measured (b) frequency responses for the fabricated C-band filter of Figure 4.8. (Source: Reprinted with permission from [30]; copyright 2004, IEEE.)


CSRRs combined with series gaps, whereas CSRRs combined with shunt stubs areused for the forward sections [33]. The lumped-element equivalent circuit modelfor the CSRR/gap stage was presented in Chapter 3 (Fig. 3.38). The equivalentcircuit model for the right-handed CSRR/stub cell is depicted in Figure 4.14.L and C are the inductance and capacitance of the line, respectively, and Lpmodels the inductance of the shunt stubs. As in Chapter 3, CSRRs are modeled asresonant tanks electrically coupled to the line through the capacitance C. Thecircuit model corresponding to the CSRR/stub stage is valid as long as CSRRs areelectrically small (as in Chapter 3, coupling between neighbor CSRRs is neglected).The dispersion relation corresponding to an infinitely long structure composed of theunit cells depicted in Figure 4.14 is given by

cos blð Þ ¼ 1þLv2 Lp þ

Lc1� LcCcv2

� �� L

C

2LcLpv2

1� LcCcv2� Lp

C

� � : (4:2)

FIGURE 4.12 (a) Layout of the fabricated S-band filter drawn to scale. The dimensions ofthe SRR pairs corresponding to the strip and gap stages are r ¼ 5.2 mm, c ¼ 0.44 mm, and d ¼0.22 mm, and r ¼ 4.1 mm, c ¼ d ¼ 0.55 mm, respectively. Strip and gap dimensions havebeen set to ww ¼ 2.8 mm and lg ¼ 6.6 mm, respectively. With these geometries, the equivalentinductance and capacitance of the strip and gap have been estimated to be Lp ¼ 183 1pH andCg ¼ 147 fF, respectively (as for the C-band filter of Fig. 4.8, the parameters of the Arlon 250-LX-0193-43-11 substrate have been considered). Finally, the host CPW is a 50V line with awide strip (W ¼ 10 mm) in order to accommodate the rings. As the separation between groundplanes is wider than the dimensions of the connectors in use, it has been necessary to cascadetaper transitions at the input and output ports to allow for connector insertion. The length of theactive device region is 22.5 mm. (b) Layout of a conventional microstrip parallel coupled-linebandpass filter with comparable performance. (Source: Reprinted with permission from [32];copyright 2005, John Wiley & Sons.)


Signal propagation is only allowed in that region where the phase constant is real. Forthis, it is necessary that the shunt reactance is negative. This occurs in a narrow bandto the left of fs, the frequency that nulls the shunt impedance.9

By combining CSRR/gap stages with CSRR/stub unit cells, it is possible to syn-thesize narrow passbands with transmission zeros at both band edges. As was done inSRR-based ARLH transmission line filters, it is necessary to design the CSRR of theright-handed and left-handed stages to exhibit the transmission zeros at different

FIGURE 4.13 Simulated (a) and measured (b) frequency responses for the fabricatedS-band filter. Insertion losses (simulated) for the conventional three-stage coupled-linebandpass filter are depicted in gray. (Source: Reprinted with permission from [32]; copyright2005, John Wiley & Sons.)

9If the inductance Lp were not present, the shunt impedance would be negative from the origin up to fs.Thus, the inductance Lp is introduced to generate a limited forward band.


frequencies, and to design either filter stage with identical central frequency.Illustrative examples of narrow microstrip bandpass filters based on ARLH linesimplemented by means of CSRRs are given in [33].

These ARLH transmission-line bandpass filters can be applied to the synthesis ofmicrowave diplexers [34]. A diplexer is a three-port device that is used at the inputstage of communication transceiver front-ends to separate the receiver (Rx) and trans-mitter (Tx) channel signals. To this end, a Rx and a Tx filter is required. A possibleconfiguration for the diplexer is that shown in Figure 4.15, where the Rx and Tx filtersare cascaded at the output ports of a Y-junction. The main relevant parameters repre-sentative of diplexer performance are in-band losses for the Tx and Rx channels(which should be as small as possible) and Rx/Tx isolation (which should be highto avoid interfering signals between the Rx and Tx channels). A prototype deviceoperative in the 2.4–3.0 GHz frequency band is depicted in Figure 4.16a. The Txand Rx filters (implemented by means of CSRR-based ARLH microstrip lines10)have been designed to provide passbands centered at 2.4 GHz and 3.0 GHz, respect-ively, with absolute bandwidths of 0.25 GHz (namely 10.3% and 8% fractional band-widths for transmission and reception). Figure 4.16b depicts the measuredtransmission coefficient for the Tx and Rx filters (i.e., S21 and S31, respectively), as

FIGURE 4.14 Equivalent T-circuit model for the CSRR/stub cell.

FIGURE 4.15 Structure of the diplexer.

10Notice that in the right-handed sections of the Rx and Tx filters, two CSRRs are used. The reason for thisis the moderate bandwidth (rather than narrow) of either filter. To achieve such bandwidths, it is necessaryto implement unit cells also with moderate 3 dB bandwidths. For the forward sections, this is more easilyachieved by using two CSRRs, rather than only one.


well as the measured Rx/Tx isolation (S32). In-band losses lower than 2 dB aremeasured for either filter, and return losses (also represented in Fig. 4.16b) arebetter than 10 dB. The frequency response of the filters is quite symmetric and themeasured isolation between ports 2 and 3 is in the vicinity of 40 dB. Remarkablealso are the dimensions of the diplexer (see the region indicated in Fig. 4.16a),which are as small as 29.8 mm � 16.3 mm (namely 0.63l � 0.34l, l being thesignal wavelength at the Tx frequency) thanks to the compact resonators used.

4.2.4 CSRR-Based Bandpass Filters with Controllable Characteristics

In this section, a methodology for the design of bandpass filters with controllablecharacteristics in microstrip technology is proposed. It is based on cascading filterstages consisting of a combination of CSRRs, series capacitive gaps, and groundedstubs. By this means we achieve the necessary flexibility to simultaneously obtainquite symmetric frequency responses, controllable bandwidths, and compact

FIGURE 4.16 (a) Topology of the fabricated ARLH microstrip line diplexer and (b) themeasured frequency response. The upper metal level is depicted in gray, and the lowermetal is drawn in black. Gap spacing is 1.33 mm and 1.67 mm for the Rx and Tx filters,respectively, and the shunt strip dimensions are 4.54 mm � 11.9 mm and 5.70 mm � 14.48mm. External CSRR radii are 2.18 mm, 2.73 mm, 2.56 mm, and 3.27 mm for the Rx (1ststage), Rx (2nd stage), Tx (1st stage), and Tx (2nd stage), respectively. The structure hasbeen implemented in the Rogers RO3010 substrate (thickness h ¼ 1.27 mm, dielectric constant1r ¼ 10.2). (Source: Reprinted with permission from [34].)


dimensions. Compact dimensions and symmetric frequency responses were obtainedby means of the ARLH concept proposed in the previous subsection. However, thesynthesis of bandpass filters with controllable bandwidth (over wide margins) wasnot considered. It will be shown in this section that by means of the proposedapproach we can obtain wide- and even ultrawideband (UWB) frequency responses.Moreover, by sacrificing periodicity, it is possible to synthesize standard frequencyresponse filters (Butterworth, Chebyshev, and so on), as will be shown later.Another interesting aspect to highlight is that the basic cell of the structure exhibitsa transmission zero above the pass band of interest. This transmission zero can be tai-lored to some extent and hence it can be useful to improve the out-of-band perform-ance of the filters through suppression of undesired harmonic bands. We will alsoconsider in this section alternative topologies for the basic cell, where the seriesgaps are replaced with transmission line sections. Although in this case wave propa-gation is forward, we can also take the benefit of the small dimensions of the resona-tors (CSRRs) to achieve compact designs. It will be shown that this latter strategy ispowerful for the synthesis of UWB response filters.

4.2.4.1 Bandpass Filters Based on the Hybrid Approach: DesignMethodology and Illustrative Examples The CSRR-based microstrip filters pro-posed in this section are planar structures that can be modeled by the circuit ofFigure 4.17a, which consists of a cascade of admittance inverters (with normalizedadmittance J ¼ 1) alternating with shunt-connected resonators tuned at the centralfrequency of the filter band, fo [35,36]. This circuit is inferred from the lowpassfilter prototype by the well-known frequency and element transformations that canbe found in any microwave textbook [18] or in monographs devoted to microwavefilters [35,36].11 By properly designing the shunt resonators, the synthesis of standardfrequency approximations is potentially possible. Infact, the transformation from thelowpass filter prototype leads to the structure of Figure 4.17b, with parallel LC res-onant tanks. Thus, as long as the resonator’s admittances fit those of the LC tanks(which depend on the L and C values inferred from circuit transformation), the tar-geted approximation (Butterworth, Chebyshev, and so on) is achievable. Generallyspeaking, in planar circuit technology, the resonators can be implemented either bymeans of a distributed approach, or by using planar semi-lumped elements.Therefore, the ideal case of perfect fitting at all frequencies is not possible, and theadmittances of the LC resonators can only be approximated near resonance. This isthe conventional procedure for the design of filters with limited bandwidth, andthis will be the case considered in this section.

The elemental cell of the proposed CSRR-based filters is depicted in Figure 4.18[37]. This consists of a CSRR etched in the ground plane (underneath the conductorstrip), combined with two series gaps and two shunt-connected metallic strips (stubs),which are grounded by means of vias. The equivalent-circuit model of this basic cellis also shown in Figure 4.18. CSRRs are modeled by means of an LC resonant tank

11Alternatively, bandpass filters can be synthesized by cascading impedance inverters and series-connected resonators.


(Lc and Cc), and their coupling to the line is described by means of the capacitance C,which depends on the portion of the inter-metallic region between the series gaps thatlies face-to-face with the metal inside the inner slot of the CSRR. The grounded stubsare modeled by means of a shunt inductance, Lp, and Cg accounts for the series gaps.The elements of the basic cell, including the CSRR, are electrically small, this being anecessary condition to properly describe the cell by means of the proposed lumped-

n

FIGURE 4.17 Generalized bandpass filter network with admittance inverters and shuntresonators (a). In (b) the resonators are LC resonant tanks.

FIGURE 4.18 (a) Basic cell of the proposed filters, (b) the equivalent-circuit model, and (c)The T-model of the elemental cell. The upper metallization is depicted in black, and the bottommetal regions are depicted in gray.


element equivalent circuit. The series inductance of the line is neglected in this study,as was done in Chapter 3 in the modeling of microstrip lines loaded with CSRRs andseries gaps. Indeed, the basic cell shown in Figure 4.18 is inspired by the elementalcell of the left-handed microstrip line depicted in Figure 3.36. The introduction of theshunt-connected stubs in the structure is a consequence of the need to improve theupper transition band of the filter. However, the introduction of such elements rep-resents an additional degree of freedom that is relevant not only to improve frequencyselectivity, but also to ease the design and to improve the out-of-band rejection of thefilter, as will be shown later.

Inspection to the circuit of Figure 4.18b reveals that the electrical model of theproposed unit cell is a combination of the dual transmission-line model and theresonant-type model. Therefore, this model can be called the hybrid model, andthe methodology based on the left-handed unit cells depicted in Figure 4.18a canbe identified as the hybrid approach [38]. Another key advantage of the structureof Figure 4.18 is related to the admittance inverter. Because the characteristicadmittance of the inverters is unity (Fig. 4.17), the inverters merely act as 908transmission lines. However, because gap capacitances are present in the structures,they can be used to obtain the desired phase shift, without the need to physicallycascade 908 transmission lines between the resonators, something relevant todevice miniaturization.12

The design methodology of the filters consists in determining the electricalparameters of the equivalent-circuit model for either filter section (provided thefilter is not periodic) from given specifications (central frequency, fractionalbandwidth, and filter order), and also the synthesis of the layout of the structure.Let us first consider the first aspect. In the circuit of Figure 4.17b, the central filterfrequency, fo, is determined by the resonance frequency of the shunt LC tank,which nulls its admittance. However, this is not exactly the case for the filterimplemented by cascading the elemental cells depicted in Figure 4.18, because theresonators and admittance inverters are intermixed. Specifically, fo, neither coincideswith the intrinsic resonance of the CSRRs, nor does it null Yp(v) ¼ Zp

21(v).Nevertheless, at fo, the phase shift and transmission coefficient between the inputand output ports of the basic cell should be f ¼ 908 and jS21j ¼ 1, respectively.This means that at fo the image impedance (or Bloch impedance for a periodic struc-ture), ZB, should coincide with the reference impedance of the ports, which is usuallyset to Zo ¼ 50V. If we now consider that the circuit of Figure 4.18b can be describedby means of its T-circuit model, with series impedance Zs and shunt impedance Zp(Fig. 4.18c), and we take into account that the phase shift and image impedancefor this circuit are given by expressions (3.1) (with f ¼ bl ) and (3.2), respectively,the previous conditions lead us to Zs ¼ 2jZo and Zp ¼ jZo. In fact, the dual solution(Zs ¼ jZo and Zp ¼ 2jZo) also satisfies the previous conditions on phase shift andimpedance matching, but this solution is not compatible with the circuit ofFigure 4.18b, because the series impedance of this circuit is capacitive.

12Nevertheless, in other filter implementations that will be analyzed later, such admittance inverters will beimplemented by means of transmission line sections.


Consequently, at the central filter frequency, the series reactance is negative (capaci-tive), and the shunt reactance (corresponding to the parallel combination of Lp and theimpedance of the CSRRs coupled to the line) is positive and, hence, inductive.According to the signs of these reactances, the filter structure composed of theelemental cells depicted in Figure 4.18 supports backward waves and behaves as aleft-handed transmission line. To determine the element values for the circuit ofFigure 4.18b, the series and shunt impedances have to be set to Zs ¼ 2jZo andZp ¼ jZo, respectively, at fo. This does not univocally determine the element valuesfor the shunt impedance. These values are also determined by the 3 dB bandwidthof the resonators,

D ¼ v2 � v1

vo(4:3)

(where vo, v1, and v2 are the central—angular—frequency and 3 dB frequencies,respectively), and by the transmission zero, which occurs at that frequency wherethe shunt impedance reduces to zero, namely

fZ ¼ 1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLc(Cc þ C)

p (4:4)

For an LC parallel resonant tank, with inductance and capacitance Leq and Ceq,respectively, we have

D ¼ 2Zo

ffiffiffiffiffiffiffiLeqCeq

s(4:5)

If we consider the filter structure of Figure 4.17b, where the L and C valuescome from the low-pass filter prototype by frequency and element transformationaccording to [35]:

Ceq ¼1

FWB � vo � Zo

� �gi (4:6)

Leq ¼1

v2oCeq

(4:7)

and expressions (4.6) and (4.7) are introduced in equation (4.5), the followingexpression for resonator bandwidths results:

Di ¼2FBWgi

(4:8)


where the gi factors are the element values of the lowpass filter prototype (whichdepend on the specific filter approximation and can be inferred from publishedtables) and Di is the 3 dB bandwidth of the resonators. According to expression(4.8), Di is proportional to the fractional bandwidth, FBW, and hence it is determinedfrom the required bandwidth of the filter. Obviously, Di is also dependent on gi, and itis therefore determined by the type of response and order. Once Di are known, we canforce the frequencies v1 and v2 to be equidistant from the central filter frequency. Atthese frequencies, under the assumption that Zs(v) does not substantially vary alongthe passband, the shunt impedance of the unit cell becomes Zp ¼ jZo/2 and infinity,respectively (see Problem 4.1), and, as has been previously indicated, Zp ¼ jZo at vo.These conditions can be expressed as

LpLcv31(C þ Cc)� Lpv1

Lcv21(C þ Cc)� Cv2

1Lp(LcCcv21 � 1)� 1

¼ Zo2

(4:9)


2Lp(LcCcv22 � 1)� 1 ¼ 0 (4:10)

LpLcv3o(C þ Cc)� Lpvo

Lcv2o(C þ Cc)� Cv2

oLp(LcCcv2o � 1)� 1

¼ Zo: (4:11)

The previous approximation (which is valid for narrow and moderate bandwidths)leads us to simple analytical expressions (equations (4.9) and (4.10)). If this approxi-mation is not applied, then the conditions arising from the 3 dB frequencies are notmathematically simple. Solution of equations (4.9)–(4.11) and (4.4) leads us to theparameters of the shunt reactance, and the series capacitance is given by

Cg ¼1

2Zovo: (4:12)

The criterion to set the transmission zero frequency, fz, obeys a compromisebetween the need to obtain a sharp transition in the upper band edge, and the conveni-ence to separate the spurious responses as much as possible from the passband ofinterest, and thus optimize the out-of-band performance of the filter. Namely, anarrow spurious band above the passband of interest arises, unless fz is properly set.This spurious response is due to the presence of a parasitic half-wavelength resonatorcoupled by the series capacitances (gaps) of the unit cells. By adjusting fz to the pos-ition of the frequency parasitic, this spurious response can be minimized, and the filterfrequency response can be significantly improved. However, the position of the spur-ious response is not known a priori. So the transmission zero frequencymust be set to acertain tentative value, for instance fz ¼ 2fo, which is expected to lie in the vicinity ofthe spurious band.13 From the previous equations, the parameters of Figure 4.18b are

13There is no physical reason that justifies this choice. However, in practice, the typical electrical sizes ofCSRRs makes the spurious band appear in the vicinity of the first harmonic of the central filter frequency.This is not something absolute, but it has been corroborated by several examples. Nevertheless, the locationof the spurious band can be estimated from the distance between the series gaps of adjacent cells.


inferred, and the topology of each filter cell (Fig. 4.18a) is obtained. The modeldescribed in [39] can be used to obtain an initial guess for CSRR dimensions (the val-idity of the model is subject to conditions that do not exactly apply in the consideredfilter cells [39]). The coupling capacitance, C, can be adjusted by partially removingthe metal delimited by the CSRR contour. The length and width of the groundedmetal strips (stubs) can be determined from independent full-wave electromagneticsimulations carried out in microstrip transmission lines loaded with these elements,or by using standard formulas [18]. A similar procedure can be used to determinethe geometry of the series capacitances. From this initial layout, the frequencyresponse of the complete filter is then simulated by means of a full-wave electromag-netic solver, thus making visible the position of the spurious band. To eliminate thisband, fz must be forced to coincide with the center of the spurious band, and themodel parameters must be recalculated. To determine the final layout, cell dimensionsshould be adjusted (starting from the seeding topology) in order to fit as closely aspossible the electrical response obtained from the latter model parameters. In practicethis is simple, because cell bandwidth is mainly controlled by Lp and C (provided fzand fo are distant enough). Therefore, stub dimensions and the etched area inside theCSRRs can be adjusted to match the required bandwidth, and the transmission zerofrequency can be tailored by scaling the CSRR dimensions.

Let us now consider two illustrative examples of bandpass filters designed by fol-lowing this approach. The first example is a bandpass filter based on a periodic rep-etition of the basic cell. The second example is a bandpass filter where periodicity issacrificed in favor of the synthesis of a standard (Chebyshev) frequency response. Inboth cases, significant levels of miniaturization are achieved, as compared to conven-tional distributed implementations, as will be shown later. The specifications of theperiodic filter are as follows: order 3, central frequency fo ¼ 1 GHz and fractionalbandwidth FBW ¼ 10%. This filter does not obey any standard approximation, there-fore we cannot directly determine gi from tabulated values, as is usual. Obviously,due to periodicity, all resonators must have the same D, and accordingly the sameg. To obtain the value of g, an order-3 low-pass filter prototype with identicalelement values has been considered, and it has been forced to exhibit the 3 dBcutoff at the normalized v ¼ 1 rad/s angular frequency. From this, g ¼ 1.521 isobtained; hence the 3 dB bandwidth of the resonators, D, is perfectly determined.From the previously explained procedure, the element values of the equivalentcircuit model have been obtained. They are identical for the three filter cells andare depicted in Table 4.1. Layout dimensions for the basic cell are indicated inFigure 4.19 (the parameters of the Rogers RO3010 substrate have been used; thick-ness h ¼ 1.27 mm, dielectric constant 1r ¼ 10.2).

TABLE 4.1 Element Values of the Equivalent-CircuitModel for the Filter of Figure 4.20

Cg (pF) Lp (nH) C (pF) Cc (pF) Lc (nH)

1.59 1.33 12.33 21.7 0.23


The simulated (using Agilent momentum) frequency response (amplitude andphase) of the single-cell structure is represented in Figure 4.19 and compared tothat obtained from the electrical simulation (insertion losses only) of the equivalent-circuit model (with the parameters indicated in Table 4.1). A reasonable fit betweenthe electrical and the electromagnetic simulations has been obtained. The simulatedand measured insertion and return losses for the fabricated three-stage filter aredepicted in Figure 4.20. Thanks to the transmission zero, the frequency response is

FIGURE 4.19 Simulated frequency response (amplitude and phase) corresponding to thebasic filter cell of Figure 4.18a. The insertion losses obtained by electrical simulation of theequivalent circuit model are also depicted (thin line). Dimensions are indicated in (b). The posi-tive phase (þ908) of S21, obtained at fo, clearly points out the left, handed nature of the struc-ture. (Source: Reprinted with permission from [37]; copyright 2006, IEEE.)


spurious free up to approximately 3fo (this transmission zero is clearly visible inFig. 4.19, where it is not obscured by parasitic resonances, as only one stage hasbeen considered in that figure). In-band insertion and return losses are good (i.e.,IL, 1.5 dB and RL . 17 dB), and frequency selectivity at both band edgesis high, with near-symmetric transition bands. The measured fractional bandwidth

FIGURE 4.20 Layout of the fabricated periodic filter (a), and simulated (b) and measured(c) insertion and return losses. Total device length excluding access lines is 4.56 cm. In (b),the simulated insertion losses for a conventional order-3 coupled-line filter with similarperformance are depicted (dashed line). (Source: Reprinted with permission from [37];copyright 2006, IEEE.)


is FBW ¼ 8%, which coincides to a good approximation with the nominal value (thediscrepancy is due to the narrow band approximation related to the nonideal admit-tance inverters used). Cell dimensions (length) are small compared with signal wave-length at fo (i.e., lc � l/7). Further miniaturization can be achieved with the penaltyof critical (smaller) dimensions being closer to the limits imposed by the fabricationtechnology (�100 mm). To avoid problems related to fabrication tolerances, criticaldimensions substantially larger than 100mm have been considered in this design,with the result of a moderately small cell size. Figure 4.20b also includes the simu-lated frequency response (insertion losses) obtained on a conventional microstripcoupled-line bandpass filter with similar performance (layout comparison is shownin the inset of Fig. 4.20b). As compared to the conventional response, where aspurious band is present at 2fo, the CSRR-based prototype device filter exhibitsnear-40 dB rejection at that frequency, and the device length, excluding accesslines, is 2.4 times shorter. Obviously, the size of the conventional device can bepartially improved by folding the coupled half-wavelength resonators in a U-shape.However, the resonator’s length cannot be reduced in contrast to our basic cells,where the limits of miniaturization are given by technological constraints, ratherthan by the signal frequency. This prototype device, published in [37], is the firstleft-handed transmission line based on CSRRs (hybrid approach) used for thedesign of a bandpass filter following a methodology able to provide the elementvalues of the equivalent-circuit model.

The nonperiodic structure is a third-order Chebyshev bandpass filter with 0.3 dBripple and 9% fractional bandwidth, centered at fo ¼ 2.5 GHz. In this case, filterstages are not identical on account of the different element values, gi, of the low-pass filter prototype. These have been inferred from table values corresponding tothe considered ripple and, from expression (4.8), the 3 dB bandwidth, Di, for eachresonator has been obtained. From these values, circuit parameters for either filtersection have been calculated by means of expressions (4.9)–(4.11) and (4.4), andthe seeding filter layout has been inferred as explained before. By forcing the positionof the transmission zeros at the convenient values, element parameters have beenrecalculated (see Table 4.2) and the optimized filter layout has been inferred asexplained previously (see Fig. 4.21). In this case, two transmission zeros at 4 GHzand 5 GHz have been considered to reject the spurious band. This prototype devicehas been fabricated on a Rogers RO3010 substrate with dielectric constant 1r ¼10.2, but thinner dielectric layer (i.e., h ¼ 0.635 mm). The measured frequencyresponse is also depicted in Figure 4.21 and compared to that obtained from the

TABLE 4.2 Element Values for the Equivalent-Circuit Model of the Filter ofFigure 4.21

Filter Cell Cg (pF) Lp (nH) C (pF) Cc (pF) Lc (nH)

1 0.63 0.49 6.11 20.50 0.0592 0.63 0.67 3.77 1.03 0.2173 0.63 0.49 6.11 20.50 0.059


equivalent circuit model (by electrical simulation using Agilent ADS). It has also beencompared to the simulated frequency response inferred on the circuit model obtainedby frequency and element transformation from the lowpass filter prototype. Verygood agreement between the electrical simulation of the equivalent-circuit modelof the filter and the circuit that results from transformation of the lowpass filterprototype has been found in the region of interest. This indicates that the lumped-element model of Figure 4.18b can model, to a very good approximation, any stan-dard approximation inferred form the lowpass filter prototype. The slight discrepancy

FIGURE 4.21 Layout and relevant dimensions of the nonperiodic CSRR-based Chebyshevbandpass filter (a) and measured insertion (bold solid line) and return (bold dashed line) losses(b). The insertion losses obtained from circuit simulation of the equivalent circuit model of thefilter (thin line) and from the low-pass filter prototype transformation (dotted line) are alsodepicted for comparison. (Source: Reprinted with permission from [37]; copyright 2006 IEEE.)


between the measured filter fractional bandwidth FBW ¼ 7% and the nominal valuehas been explained before with reference to the previous periodic filter implemen-tation. Moreover, it has to be taken into account that losses have been neglected inthe electrical simulations. Measured in-band insertion and return losses are IL ¼

1.9 dBs and RL . 13 dB, respectively. Finally the authors want to highlight againthe small dimensions of the device, with a length, excluding access lines, of l ¼2l/5 (l being the signal wavelength at the central filter frequency). As comparedto conventional distributed approaches, the design methodology presented in thissection makes it possible to simultaneously achieve small dimensions, low losses,high-frequency selectivity, and good out-of-band performance.

4.2.4.2 Other CSRR-Based Filters Implemented by Means of Right-HandedSections In the bandpass filters reported in Section 4.2.4.1, the elemental cell isdescribed by a T-circuit model where the series impedance is capacitive and theshunt impedance is inductive (positive reactance) in the passband of interest(hence the structure behaves as a left-handed transmission line section). The designmethodology for such bandpass filters, which allows for the synthesis of bandpassfilters subject to given specifications and/or standard approximations, is based onthe generalized bandpass filter network depicted in Figure 4.17. Indeed, the elementalcell depicted in Figure 4.18 is a particular case, among other possible solutions, forthe physical implementation of such a network. The phase shift per cell is positive inthe region of interest, instead of being negative, as occurs in other filter unit cellswhere the shunt resonators are coupled through 908 (actually 2908) transmissionlines acting as admittance inverters.14 This is just one of the alternative solutionsfor the synthesis of CSRR-based bandpass filters, where the characteristic impedanceof the 908 line must be set to 50V in order to obtain the desired electrical character-istics [40]. Another possibility is to replace such transmission line sections by trans-mission lines merely acting as coupling inductances [41].

Let us now analyze the synthesis of bandpass filters implemented by means ofCSRR/stub resonators15 coupled by means of l/4 (or 908) lines. The topology ofthe basic cell is depicted in Figure 4.22, where rectangular CSRRs have been con-sidered (the shape of the rings is not critical).16 The equivalent-circuit model isalso depicted in this figure. It is identical to that shown in Figure 4.18b, but withthe series capacitances replaced by l/8 lines with 50V characteristic impedance.By cascading such filter sections, the resulting structure can be described by thegeneralized bandpass filter network shown in Figure 4.17. The admittance inverter(l/4 lines) and the shunt resonator (formed by the parallel combination of thegrounded stubs and the capacitively coupled, through C, CSRRs) can be perfectlyidentified. The design equations are similar to those reported in Section 4.2.4.1,

14The positive phase shift per cell (phase of S21) in the structure of Figure 4.18 (see Fig. 4.19a) correspondsto a negative phase constant in the region of interest (left-handed passband).15By CSRR/stub resonators the authors mean those shunt resonators of the elemental unit cell of the hybridapproach, namely, consisting of a CSRR (coupled to the line through a capacitance C ) combined with ashunt stub.16In the topology of Figure 4.22, only one grounded stub is considered. Obviously, two stubs are also poss-ible, the circuit model of the unit cell being identical.


but they are not identical. Namely, the 3 dB bandwidth of each resonator is given byexpression (4.8). To univocally determine the four electrical parameters of theequivalent circuit model, four independent equations are required. One of suchequations is given by the transmission zero frequency (expression 4.4). Twofurther equations arise by forcing insertion loss to be IL ¼ 3 dB at v1 and v2,where these angular frequencies are obtained from expression (4.3) and (4.8),subject to the additional constraint of being equidistant from v o. This gives



1Lp(LcCcv21 � 1)� 1

¼ Zo2

(4:13)

and



2Lp(LcCcv22 � 1)� 1

¼ � Zo2: (4:14)

Finally, the fourth equation arises by forcing the shunt impedance to be infinity at thecentral filter frequency. This gives


oLp(LcCcv2o � 1)� 1 ¼ 0: (14:15)

With these model equations, the equivalent circuit as well as the layout of the filteringstructure can be inferred (the procedure explained before can be also applied).

One prototype device example is discussed in the following steps in order to high-light the potential of this approach. Specifically, an ultrawide bandpass filter17

(UWBPF) has been designed with the following specifications: FBW ¼ 90%, fo ¼6.8 GHz, order N ¼ 3 [40]. The layout that has been obtained after optimization is

FIGURE 4.22 Layout of the CSRR/stub resonator (a) and equivalent circuit model (b). Theimpedance inverter is obtained by means of two l/8 transmission line sections (which form a908 line when several stages are cascaded to form a filter).

17UWBPFs are defined as those filters covering a bandwidth of more than 1 GHz or larger than 20%.Nevertheless, ultrawideband (UWB) communication systems are restricted to fulfill the requirements ofthe mask supplied by the Federal Communication Commission (FCC) in February 2002 to regulate suchsystems. This mask extends from 3.1 GHz up to 10.6 GHz. Therefore, to avoid interfering signals, filtersroughly covering this bandwidth are of interest. Usually such filters are called UWBPFs. Obviously,they satisfy the above definition, but these UWBPFs do not represent the exclusive definition of the term.


depicted in Figure 4.23 (the relevant dimensions are indicated). The parameters of theRogers RO3010 substrate (dielectric constant 1 ¼ 10.2, thickness h ¼ 127 mm, havebeen considered). The simulated frequency response of this structure (obtained bymeans of Agilent momentum) is also represented in Figure 4.23. The frequency para-sitic related to the presence of an undesired inductive coupled half-wavelength reso-nator can be appreciated very close to the upper edge of the band. There is also anadditional narrow band at 17 GHz, which is due to a second-order resonance of theCSRRs. To suppress these bands, two properly tuned CSRRs are etched at thefilter output. The layout of the final filter and the simulated frequency response aredepicted in Figure 4.24a. The FBW that results from the simulated frequencyresponse is slightly lower than the nominal value. This is expected on account ofthe limited frequency range of the admittance inverters, but this effect could becompensated by over-dimensioning the FBW. In order to ease fabrication, a proto-type device scaled down in frequency has been fabricated on a Rogers RO3010substrate with identical dielectric constant (1 ¼ 10.2), but thicker dielectric layer

FIGURE 4.23 Layout of the CSRR-based filter implemented by means of CSRR/stub reso-nators and 908 impedance inverters (a) and simulated insertion and return losses (b). (Source:Reprinted with permission from [40]; copyright 2005, John Wiley & Sons.)


(h ¼ 635 mm). Lateral filter dimensions have been scaled by the same factor as thethickness, so that the lower dimensions are not so critical. The measured frequencyresponse is depicted in Figure 4.24b. Good agreement between simulation (originallayout of Fig. 4.24a) and measurements has been obtained. The measured fractionalbandwidth is FBW ¼ 87%, in-band losses (optimum value) are IL ¼ 0.3 dB, with1 dB ripple, and return losses are in the vicinity of RL ¼ 10 dB. The authorswould also like to highlight that the first spurious band arises at approximately 3fo.Therefore, the out-of-band filter behavior is quite satisfactory, with rejection levelsabove 40 dB between 2.2 GHz and 3.2 GHz and above 20 dB up to 3.7 GHz.Obviously, frequency selectivity at the lower transition band can be improved byincreasing the number of stages. To illustrate this, an eighth-order prototype-deviceperiodic filter is depicted in Figure 4.25 (the Rogers RO3010 substrate with 1r ¼

10.2, h ¼ 1.27 mm was used) [42]. The device exhibits a bandwidth covering the

FIGURE 4.24 Layout and simulated frequency response for the CSRR-based UWB filterwith improved stopband (a) and measured frequency response of the fabricated UWBPFscaled down in frequency (scaled up in dimensions) by a factor of 5 (b). (Source: Reprintedwith permission from [40]; copyright 2005, John Wiley & Sons.)


standard band for UWB applications, namely from 3.1 GHz up to 10.6 GHz. Thesimulated and measured frequency responses for this structure are shown inFigure 4.26. Good out-of-band performance is obtained, return losses in theallowed band are reasonable, and insertion losses are better than 2.4 dB. Theseresults indicate that by using CSRRs, it is possible to design very wide bandpassfilters with good out-of-band performance and small dimensions, something noteasily achievable simultaneously.

Concerning the design of bandpass filters based on CSRR/stub resonators coupledthrough transmission-line sections acting as series inductive elements, the topologyof the basic cell and its lumped-element equivalent-circuit model are both depictedin Figure 4.27 [41].18 In contrast to the previous filters, where the characteristic impe-dance of the 908 lines was always set to 50V, this impedance (and hence line width)is now an adjustable parameter that is used to achieve the required inductance value.To determine the circuit elements, we again appeal to the generalized bandpass filternetwork. At the central filter frequency, fo, the phase shift between adjacent stages isf ¼ 2908 and the image impedance is given by the reference impedance of the ports,Zo. If these conditions are applied to the T-circuit model of Figure 4.27b, the series

FIGURE 4.25 Layout (a) and photograph (b) of the fabricated eight-stage UWBPF.Relevant dimensions are indicated. (Source: Reprinted with permission from [42]; SpringerScience and Business Media.)

18Notice that the topology and circuit model of these cells are identical to those of the right-handed stagesof the ARLH filters discussed in Section 4.2.3.2.


and shunt impedances at fo are given by Zs ¼ jZo and Zp ¼ 2jZo, respectively. Notethat the signs of these impedance are the opposite to those of Section 4.2.4.1.From these impedance values, the line inductance is given by

L ¼ 2Zovo

, (4:16)

FIGURE 4.26 Simulated (a) and measured (b) frequency responses for the filter ofFigure 4.25. (Source: Reprinted with permission from [42]; Springer Science and BusinessMedia.)

FIGURE 4.27 Topology of the basic cell for the filters based on inductive coupling, andequivalent circuit model.


and the following condition for the elements of the shunt reactance must hold:

Lpvo � LpLcv3o(C þ Cc)


oLp(LcCcv2o � 1)� 1

¼ Zo (4:17)

The three additional equations required to univocally determine the element valuesare given by the transmission zero frequency (equation 4.4) and by the 3 dB frequen-cies, v1 and v2, of the T-circuit model. At these frequencies the shunt impedancebecomes Zp ¼ 2jZo/2 and infinity, respectively. Therefore,

Lpv1 � LpLcv31(C þ Cc)

Lcv21(Cc þ C)� Cv2

1Lp(LcCcv21 � 1)� 1

¼ Zo2

(4:18)

and


2Lp(LcCcv22 � 1)� 1 ¼ 0, (4:19)

and v1 and v2 are determined from equations (4.3) and (4.8). These equations arevery similar to those reported in Section 4.2.4.1. However, the fact that the shuntimpedance is capacitive in the region of interest, makes the sign of expression(4.18) to be opposite to that of equation (4.9). The design procedure for thesefilters is very similar to that explained in the previous subsection. To determine thewidth of the transmission-line sections between adjacent resonators, the line induc-tance is forced to take the value given by expression (4.16). This inductance is iden-tical for all the filter stages. However, because filter section lengths may not beidentical (due to the variable CSRR size of the different stages), slightly narrowerlines are required in those sections with smaller etched rings.

A prototype-device bandpass filter representative of this latter approach is given inFigure 4.28 [41], together with the required specifications (further details on the syn-thesis of this filter can be found in [41]). To achieve the required 3 dB bandwidths foreither resonator section, it has been necessary to use double-slit CSRRs (2-CSRRs),and it has also been necessary to etch two 2-CSRRs pairs at the input and outputstages of the filter. The simulated and measured frequency responses of the device,also depicted in Figure 4.28, show that the target specifications are satisfied. Themeasured central frequency and fractional bandwidth are fo ¼ 3.8 GHz and 31.2%,respectively. Measured in-band losses are below 1 dB between 3.36 GHz and 4.15GHz, with an optimum value of 0.7 dB at fo. Return losses better than 15 dB havebeen obtained between 3.37 GHz and 4.27 GHz. Finally, the out-of-band filter per-formance is good, with 40 dB rejection at the desired frequency regions. The sizeof the filter, excluding access lines, is as small as 30 mm � 16 mm.

The bandpass filters studied in this section can be described by the network rep-resentation depicted in Figure 4.17, consisting of a cascade of shunt resonators andadmittance inverters. However, bandpass responses can also be synthesized by cas-cading series resonators and impedance inverters. As open split rings resonators(OSRRs; see Chapter 2 and Problem 2.6) are modeled as series resonant tanks[43], bandpass filters based on these particles can also be designed. The design


methodology is very similar to that corresponding to CSRR/stub shunt resonatorscoupled through 908 transmission lines and, hence, it is not reproduced here(details of such filters can be found in [44]).

4.2.5 Highpass Filters and Ultrawide Bandpass Filters (UWBPFs)Implemented by Means of Resonant-Type Balanced CRLH MetamaterialTransmission Lines

It was pointed out in Chapter 3 that left-handed lines implemented by means of SRRsor CSRRs exhibit a CRLH behavior, and that by properly designing such lines it ispossible to collapse the gap between the left-handed and right-handed bands

FIGURE 4.28 (a) Layout of a filter implemented by means of 2-CSRR/stub resonatorscoupled through inductive transmission lines, and relevant dimensions. (b) Target specifica-tions (insertion loss) for the filter prototype (gray regions represent forbidden loci). (c)Simulated (dashed line) and measured (solid line) insertion and return losses for the fabricatedprototype filter. For the external 2-CSRRs, rings slot and separation are 0.29 mm, and for the2nd, 3rd and 4th stages, these are 0.30 mm. The device has been fabricated on a RogersRO3010 substrate with dielectric constant 1r ¼ 10.2 and thickness h ¼ 1.27 mm. (Source:Reprinted with permission from [41]; copyright 2005, John Wiley & Sons.)


(balanced design). Balanced CRLH lines exhibit typically broad bandwidths that areuseful for the synthesis of highpass filters (HPFs) or ultrawide bandpass filters(UWBPFs). In microwave engineering the difference between an HPF and aUWBPF is subtle. Namely, a microwave HPF implemented in planar technologydoes not infact exhibit an unlimited (i.e., extending up to infinity) band above thecutoff frequency. Thus, microwave HPFs are indeed bandpass filters with a verywide band and a controllable cutoff frequency at the lower band edge. In otherwords, the difference between an HPF and a UWBPF is merely the fact that inthese latter filters, both the upper and lower cutoff frequencies are controllable, andin HPFs the upper limit of the band is not a target specification. With these commentsin mind, it is clear that balanced CRLH CSRR-based unit cells such as those depictedin Figure 3.46 are useful for the synthesis of HPFs. They exhibit a huge transmissionband above the cutoff, and frequency selectivity is good due to the presence of atransmission zero. Moreover, rejection below cutoff can be controlled by thenumber of stages of the structure. To illustrate this possibility, a prototype-deviceHPF consisting of three stages, identical to that depicted in Figure 3.46 has beenfabricated and characterized (Fig. 4.29) [45]. For comparison, also depicted in thisfigure are the simulated frequency responses of two- and four-stage devices.As has been indicated, rejection below cutoff is controlled by the number of cells.

FIGURE 4.29 Layout of the microstrip filter formed by three balanced CRLH CSRR-basedcells (a) and measured frequency response (b). In (b) are also depicted the simulated insertionand return losses for two-, three-, and four-stage device filters. The metallic parts are depictedin black in the top layer and in gray in the bottom layer. Dimensions are: total length l ¼ 55mm,line width W ¼ 0.8 mm, external radius of the outer rings r ¼ 7.3 mm, ring width c ¼ 0.4 mmand ring separation d ¼ 0.2 mm; the interdigital capacitors are formed by 28 fingers separatedby 0.16 mm. (Source: Reprinted with permission from [45]; copyright 2007, IEEE.)


For the measured three-stage device filter, rejection below cutoff is better than 50 dB,and in-band insertion and return losses are good up to 3 GHz.

To synthesize a UWBPF by using the previous structures, it is necessary to includeadditional elements able to limit the transmission band. These elements can also bemultituned CSRRs (see Section 4.2.1), acting as negative-permittivity particles anddesigned to achieve the upper filter cutoff at the required frequency. In Figure 4.30a UWBPF is depicted roughly covering the mask supplied by the FederalCommunication Commission (FCC) to regulate ultrawideband (UWB) communi-cation systems (3.1–10.6 GHz). In this filter, the CSRRs introduced to control theupper stopband are etched inside those CSRRs that are used to synthesize thebalanced CRLH structure. Moreover, additional CSRs (complementary spiral resona-tors) can be seen cascaded to the input and output ports. These latter elements areintroduced to demonstrate that it is possible to provide a transmission zero withinthe allowed band, this being of interest for the rejection of potential interferingsignals, such as WLAN, among others19 [46,47]. The simulated and measuredfrequency responses of this filter are depicted in Figure 4.30b.20 As expected, anotch in the vicinity of 5 GHz results due to the presence of the CSRs. The measuredbandwidth roughly coincides with that given by the FCC (discrepancy betweenmeasurement and simulation is attributed to fabrication-related tolerances). OtherUWBPFs based on a combination of CSRRs and SRRs covering the FCC maskhave been proposed [48]. Broadband filters based on balanced dual transmissionlines can be found in [49].

4.2.6 Tunable Filters Based on Varactor-Loaded SplitRings Resonators (VLSRRs)

Reconfigurable (tunable) devices are required in modern communications systemsthat need to adapt to changing operating conditions. Among these devices, filters areprobably the microwave components where most efforts have been devoted towardsfinding solutions providing accurate control of their electrical characteristics andtunability over wide ranges. Tunable filters are based on electronically controllablecomponents such as varactor diodes or RF-MEMS (microelectromechanical switches),among others, where a variable capacitance is used to tailor device characteristics.Varactor diodes and RF-MEMS have been applied to the design of tunable devices,such as filters and other microwave components, based on conventional configurations[50]. The possibility of designing tunable filters and resonators bymeans of active elec-tromagnetic bandgaps (EBGs) [51] has also been demonstrated based on varactordiodes. In this section, it is demonstrated that tunable filters can be implemented byusing resonant-type metamaterial transmission lines loaded with varactor diodes.

19For the designed structure, the transmission zero in the vicinity of 5 GHz is of interest for the eliminationof WLAN radio signals following the IEEE 802.11a or Hiperlan/2 standards (data rates up to 54Mbps in aregulated frequency spectrum around 5 GHz).20Although UWBPF filters with better performance have been designed and fabricated [48], the device ofFigure 4.30 has been chosen to illustrate the possibility of etching the smaller CSRRs (responsible for theupper stopband) inside the larger CSRRs.


Though not exclusive, one possibility of achieving tunability is managed by adding thevaractor diodes to the resonant elements (SRRs). The result is the varactor-loaded splitrings resonator (VLSRR), which was introduced by Gil et al. [52]. By loading a trans-mission line with VLSRRs and biasing the varactors at different voltages, the trans-mission characteristics of the structure can be tailored.

4.2.6.1 Topology of the VLSRR and Equivalent-Circuit Model The originaltopology (layout) of the VLSRRs proposed by Gil et al. [52] is depicted inFigure 4.31a. It is similar to the topology originally proposed by Pendry [53]

FIGURE 4.30 Layout of the fabricated UWBPF (a) and frequency response (b). Dimensionsof the smaller CSRRs are c ¼ 0.17 mm, d ¼ 0.11 mm and rext ¼ 0.89 mm (those CSRRs of theextremes are slightly larger to improve the upper stopband, as is explained in the text). CSRshave been tuned to provide a transmission zero at 5 GHz (c ¼ 0.17 mm, d ¼ 0.11 mm andrext ¼ 1.01 mm). Dimensions of the larger CSRRs are rext ¼ 2.10 mm, d ¼ 0.11 mm, c ¼0.22 mm, and c ¼ 0.16 mm for the external and internal ring, respectively. The area of thedevice (dashed rectangle) is A ¼ 1.77 cm � 0.41 cm. CSRRs and CSRs are printed in theground plane (in gray). The Rogers RO3010 substrate with thickness h ¼ 635 mm and dielec-tric constant 1r ¼ 10.2 has been used.

FIGURE 4.31 Topologies of the tunable VLSRR (a) and VLSRR with improved geometry(b). The relevant dimensions are indicated. (Source: Reprinted with permission from [54];copyright 2006, IEEE.)


(Chapter 2), although the separation between rings, d, is no longer uniform in order toconnect the diode varactor between the internal and external conductors. A metal padis added in the center of the particle to ease diode biasing, and rectangular rings havebeen considered to enhance line-to-rings coupling. With this configuration, the elec-tromagnetic behavior of the VLSRRs does not substantially differ from that of theSRR, except in the fact that certain electronic control of the resonance frequency ispossible thanks to the varactors, which are connected between the inner and outerconductors and dominate over the edge capacitance corresponding to the right halfof the structure. Another difference between SRRs and VLSRRs concerns their exci-tation, or the generation of current loops at resonance. In the former, these currents aremainly induced by the magnetic field flowing into the inner ring, whereas inVLSRRs, rings excitation is mainly achieved by the magnetic field penetrating theinter-rings region where the varactors are allocated. Alternatively, the topology pre-sented in Figure 4.31b, can also be considered to achieve tuning [54]. As comparedto the structure of Figure 4.31a, the right-hand arm of the outer ring has been shor-tened because no appreciable current flows through it (this has been corroborated byelectromagnetic simulation). In other words, the electric current is mostly absorbedby the varactor diode, preventing it from circulating across the portion of the externalring between the diode junction and the slit.

The lumped-element equivalent-circuit model of a biased VLSRR coupled to amicrostrip transmission line is depicted in Figure 4.32 [54]. Diode varactors aremodeled by a variable capacitance, CVAR, and a series resistance, Rs, which takesinto account not only the intrinsic losses of the diode, RD, but also the resistanceassociated with the varactor–metal junctions, RVM. CR (which is neglected due tothe shunt connection of CVAR to it) and CL are the edge capacitances correspondingto the right and left halves, respectively, of the VLSRR, and Lr and Cpad model theequivalent inductance of the VLSRRs and the pad-to-ground capacitance, respec-tively. Diode biasing is applied through a variable voltage source, Vbias, which has

FIGURE 4.32 Lumped-element equivalent-circuit model for the elemental cell of a biasedVLSRR coupled to a microstrip transmission line. L and C are the per-section inductanceand capacitance of the line. Due to symmetry, the magnetic wall concept has been used.(Source: Reprinted with permission from [54]; copyright 2006, IEEE.)


an equivalent output resistance termed Rbias. Concerning line-to-VLSRRs coupling,we a priori assume that VLSRRs can be driven either by the axial magnetic field gen-erated by the line (inductive coupling) or by the electric field present between the lineand the external ring (capacitive coupling). Both couplings have been properlymodeled by means of a mutual inductance, M (magnetic coupling), and the edgecapacitance between the line and the external ring, Cedge (electric coupling). Thecapacitances CrL and CrR are the rings-to-ground capacitances of the left and righthalves, respectively.

4.2.6.2 Validation of the Model The previous model is validated by comparingthe frequency response measured on a fabricated VLSRR-loaded microstrip line withthat obtained through electrical simulation of the lumped-element equivalent circuit.To this end, a two-stage device has been fabricated (Fig. 4.33a). BB833-InfineonTechnologies silicon tuning diodes have been used as nonlinear capacitances(the capacitance window for these diodes is 0.75–9 pF for voltages varying in theinterval 0–30 V). As expected, a stopband (due to the negative effectivepermeability of the structure) with tuning capability is obtained (Fig. 4.33b).The device has been fabricated on a Rogers RO3010 substrate (dielectric constant1r ¼ 10.2, thickness h ¼ 1.27 mm, tan d ¼ 0.0023). Except the mutual inductance,M, and the varactor–metal junction resistance, RVM, which have been used asfitting parameters, the other element values have been estimated either throughgeometrical considerations or with the help of the commercial software AgilentADS (see [54] for details). The frequency responses depicted in Figure 4.33c havebeen obtained by setting M ¼ 2.4 nH and RVM ¼ 3V (the other parameters areindicated in the caption of Fig. 4.33), and are those that have optimally fitted theexperiment over the considered tuning interval. The good agreement betweentheory and experiment supports the validity of the model.

As it is, the equivalent-circuit model includes both magnetic and electric couplingbetween the host line and the VLSRRs. However, if the electric coupling is switchedoff by removing the coupling capacitance Cedge, no appreciable differences in thesimulated frequency responses (over the tuning interval considered) come up.Moreover, no appreciable changes arise if Cpad is removed. Hence, the electriccoupling can be neglected and the equivalent circuit model of the VLSRR-loadedmicrostrip line can be simplified to that shown in Figure 4.34a, where inductiveexcitation of the VLSRRs is the single coupling mechanism (the simulated frequencyresponse with Cedge and Cpad removed is shown in Fig. 4.34b for comparisonpurposes). Indeed, this circuit can be further simplified to the circuit ofFigure 3.23b, with C 0

s replaced by Ceq ¼ CLCVAR/(CL þ CVAR) (see details inChapter 3).21

4.2.6.3 Some Illustrative Results: Tunable Notch Filters and StopbandFilters VLSRRs coupled to microstrip lines can be applied to the synthesis of

21The circuit is simplified by neglecting CrL and CrR, which are small. Thus, Rbias merely acts as a polar-ization path for the diodes (i.e., irrelevant for signal analysis).


FIGURE 4.33 Photograph of a two-stage VLSRR-loaded microstrip line (a), measuredinsertion losses for different diode polarizations (b) and frequency responses obtained by elec-trical simulation of the equivalent-circuit model shown in Figure 4.32. Relevant dimensions arec ¼ d ¼ 0.2 mm, separation between line and external rings is 0.2 mm, and length and width ofVLSRRs are 6.2 mm and 2.8 mm, respectively. Element values are Lr ¼ 3 nH, CL ¼ 1.4 pF,Cedge ¼ 1 fF, CrL ¼ 0.1 pF, Cpad ¼ 0.44 pF, CrR ¼ 0.89 pF, L ¼ 3.3 nH, C ¼ 1.3 pF, RS ¼

4.8V, (i.e., RD ¼ 1.8V and RVM ¼ 3.0V), Rbias ¼ 50V, M ¼ 2.4 nH and 1 pF � CVAR �9 pF. (Source: Reprinted with permission from [54]; copyright 2006, IEEE.)


tunable notch and stopband filters [52,54]. Indeed, the structure of Figure 4.33a is anexample of a tunable notch filter, where the notch frequency is bias-controlled.22

Rejection can be improved by adding VLSRR stages (see a four-stage device inFigure 4.35a and the corresponding response in Fig. 4.35b). By biasing the diodesat different voltage levels, it is possible to widen and to tailor filter bandwidth, asFigure 4.35c and d illustrate.23

By adding inductive vias to the previous structures, tunable bandpass filters exhi-biting left-handedness are obtained [54]. However, due to varactor diode losses, theperformance of these devices is very limited. To synthesize tunable bandpass filterswith good performance, low-loss tuning elements such as RF-MEMS are required.The combination of these elements with SRRs or CSRR is very promising.

To summarize Section 4.2, it has been shown that metamaterial transmission linesbased on SRRs and CSRRs are useful for the design of microwave filters in planartechnology. Stopband filters and bandpass filters with narrow and widebands can

FIGURE 4.34 Equivalent-circuit model of the VLSRR-loaded microstrip line with electriccoupling removed (a), and electrical simulations for different diode polarizations (b).(Source: Reprinted with permission from [54]; copyright 2006, IEEE.)

22Rejection level and bandwidth also change with tuning. This is explained by the variation in Q factor thatresults when the notch frequency is tuned (see Problem 4.2).23Another possible solution to produce wide stopbands in stopband filters is to design the VLSRR stageshaving slightly different dimensions (closed resonances are obtained) and to apply the same varying biasvoltage to the devices.


be synthesized, and a systematic design methodology has been proposed that allowsus to design filters subject to specifications, or following standard approximations.Miniaturization, optimization of device performance through elimination of spuriousfrequencies, as well as the possibility to control filter bandwidth over wide marginsare key issues. In the opinion of the authors, one of the domains where resonant-typeone-dimensional metamaterials, or structures based (or inspired) on them, can findmore practical applications is in the field of filters. However, the small dimensionsof the unit cells, as well as their nonusual electromagnetic properties, makeresonant-type metamaterials very promising in other microwave applications, aswill be explored in the next sections.

4.3 SYNTHESIS OF METAMATERIAL TRANSMISSION LINES WITHCONTROLLABLE CHARACTERISTICS AND APPLICATIONS

It has been pointed out at the beginning of this chapter that the most outstanding prop-erty of metamaterial transmission lines is the controllability of their electrical charac-teristics (impedance and phase). In these artificial lines there are more parameters totailor, compared to conventional transmission lines. This means that the number of

FIGURE 4.35 Fabricated four-stage VLSRR microstrip line (a) and measured frequencyresponse for identical bias voltage applied to the diode varactors (b). In (c) and (d ) are depictedthe measured transmission characteristics of the structure by applying the indicated voltages tothe different VLSRR stages. Dimensions are identical to those of Figure 4.33a. (Source:Reprinted with permission from [54]; copyright 2006, IEEE.)

4.3 SYNTHESIS OF METAMATERIAL TRANSMISSION LINES 233

degrees of freedom is larger and hence it is possible to synthesize transmission lineswith a combination of characteristics and dimensions not easily achievable throughconventional implementations. By engineering these lines it is also possible todesign microwave components with superior performance as compared to existingdevices, or to obtain microwave devices based on new functionalities, such asdual-band components. In this section we will review these aspects. Section 4.3.1is devoted to the miniaturization of microwave components, Section 4.3.2 isfocused on the synthesis of enhanced bandwidth devices, in Section 4.3.3 we willdiscuss the principles for the design of dual-band components, and, finally,Section 4.3.4 will be devoted to coupled-line couplers based on metamaterials.

4.3.1 Miniaturization of Microwave Components

The size reduction of microwave components is based on two features: (1) the smalldimensions of the constitutive unit cells of metamaterial transmission lines and (2) thecontrol of the electrical length of such lines, which covers a wide margin with asingle-unit cell, as was pointed out in Chapter 3.24 Planar microwave circuits basedon the distributed approach consist of combinations of transmission lines and stubswith certain characteristic impedance, ZB, and electrical length, f. Conventionaltransmission lines and stubs can thus be substituted by their metamaterial counter-parts with the possibility of significantly reducing dimensions. Obviously, the mostfavorable solution in terms of dimensions is achieved if only one unit cell is used.25

The synthesis of metamaterial transmission lines consists of the determination ofthe parameters of the circuit model of such lines from the required values of charac-teristic impedance Zn ¼ ZB( fn) and electrical length fn ¼ f( fn) at the operating(design) frequency, fn. As the example given later is based on the resonant-typeapproach (using CSRR-based left-handed lines), the considered model to infer thereactive parameters from transmission line parameters is that depicted inFigure 3.38, where the line inductance has been neglected as it is assumed that theseries impedance is dominated by the gap capacitance (this aspect was discussedin Chapter 3). From equations (3.1) and (3.2), the capacitance value of the seriesgap can be easily inferred as

Cg ¼1

2vnZn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cosfn

1� cosfn

s(4:20)

24According to the typical dispersion diagrams of left-handed and CRLH lines, the phase of the unit cellvaries between 2p and 0 along the left-handed band.25At this point we would like to mention that although the commonly accepted definition of metamaterialtransmission line refers to an effectively homogeneous periodic structure (i.e., with period much smallerthan guided wavelength) with controllable characteristics, we will be flexible and we will adopt underthis term also those structures not satisfying the former requirement (homogeneity), including structureswith a single-unit cell. The reason is that for microwave circuit design the aim is to achieve the requiredelectrical characteristics at the design frequency, rather than synthesizing an effective medium.


where vn ¼ 2pfn. To obtain the other parameters of the equivalent-circuit model,inversion of equations (3.1), (3.2), (3.56), and (3.57) is necessary. This involvesa tedious calculation. The final results are (see Problem 4.3 and [55])

Lc ¼Zn2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cosfn

1� cosfn

svn

v4H

(v2H � v2

L)(v2H � v2

n)

(v2n � v2

L), (4:21)

Cc ¼1

Lcv2H

, (4:22)

and

C ¼ 2v2H(v

2n � v2

L)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos2 (fn)

pZnvn v2

n(1þ cosfn)(v2H � v2

L)� 2v2H(v

2n � v2

L)� � , (4:23)

where vL ¼ 2pfL and vH ¼ 2pfH ( fL and fH being the lower and upper limit of theleft-handed band, given by expressions 3.56 and 3.57). Inspection of equations(4.20)–(4.23) reveals that Cg, Lc, and Cc are always positive provided vL ,

vn , vH. However, depending on the relative values of these angular frequencies,C may be negative. Therefore, the operative bandwidth of the device is limited bythe value of C, which must be a real positive number.26

To illustrate the achievable size reduction by using resonant-type left-handed lines,a Y-junction power divider is designed. In such power dividers, two 908 line admit-tance inverters are used (see the schematic in Fig. 4.36). In order to achieve

FIGURE 4.36 Schematic of a Y-junction power divider implemented by means of 908 impe-dance inverters.

26Although fL and fH are not actually design parameters, device bandwidth increases as the separationbetween these frequencies is higher. More details on the limited bandwidth of these structures is givenin [55].


impedance matching for the input port, the characteristic impedance of such 908transmission lines must be set to 1.41Zo, where Zo is the reference impedance ofthe ports (normally 50V). The metamaterial power divider can be implemented bymeans of CSRR-based left-handed lines with the required electrical parameters(Zn ¼ 1.41Zo ¼ 70.7V and fn ¼ 2908)27 at the operating frequency (which hasbeen set to fn ¼ 1.5 GHz). From these parameters and frequency, the elements ofthe equivalent-circuit model have been obtained (following the procedure describedabove). To this end, the frequencies delimiting the allowed left-handed band havebeen set to fL ¼ 1.38 GHz and fH ¼ 1.68 GHz. With these delimiting frequencies,the element values are found to be reasonable (i.e., Cg ¼ 0.75 pF, C ¼ 14.9 pF,Lc ¼ 1.76 nH, and Cc ¼ 5.08 pF). Once the element values have been inferred, thelayout of the inverter can be obtained (optimization and the parameter extractionmethod described in Chapter 3 are used). The layout of the CSRR-based powerdivider is depicted in Figure 4.37 and compared to that corresponding to a conven-tional implementation (photographs of the fabricated device are also depicted inthis figure). Significant size reduction (50% as compared to conventional implemen-tation) is achieved through the use of metamaterial admittance inverters. Themeasured and simulated frequency responses of the designed metamaterial powerdivider are depicted in Figure 4.38 (S31 is roughly identical to S21 and is notrepresented in the figure). The measured insertion losses at the design frequency(dB(S21) ¼ 23.2 dB), are close to the value corresponding to an ideal losslessdevice, and measured return losses are better than 20 dB. Thus, device miniaturiza-tion is achieved, maintaining device performance at the design frequency.Although not represented, the frequency response of the conventional device exhibitsa wider operative bandwidth. However, this does not mean that in general theoperative bandwidth of metamaterial-based devices is smaller than that achievablethrough conventional implementations. On the contrary, it will be shown in thenext section that it is possible to design enhanced bandwidth devices by usingmetamaterial transmission lines in their designs.

4.3.2 Compact Broadband Devices

Let us now try to explain the principle for the design of compact broadband micro-wave components. The limited bandwidth of microwave components is due to thephase shift experienced by transmission lines and stubs when frequency is variedfrom the nominal operating value. Let us assume that at this frequency, vn, thephase shift is f(vn) ¼ fn. This phase shift is related to the phase velocity, vp, ofthe line according to the following expression:

fn ¼ bl ¼ l

vpvn, (4:24)

27The negative sign of the phase of the left-handed admittance inverters is irrelevant for device operation.


FIGURE 4.37 Layout (a) and photographs of the top (c) and bottom (d ) faces of the CSRR-based power divider implemented by following the schematic of Figure 4.36. For comparison,the layout of a conventional power divider designed by considering an identical substrate is alsodepicted (b). The dimensions (in mm) of the two impedance inverters forming the device arethose indicated in (e). In (e), the upper metal strip is depicted in black, whereas the slot of theCSRR is depicted in gray. (Source: Reprinted with permission from [55].)


where l is the length of the line. The bandwidth of any transmission-line-based deviceis given by the frequency interval where the variation of f, namely Df ¼ f2 fn, islower than a certain predetermined value. Hence, bandwidth is intimately related tothe derivative of f with frequency, also known as group delay. In view of expression(4.24), this derivative increases with the length of the line and therefore we obtain thewell-known conclusion that bandwidth decreases with the phase shift (proportionalto l ) of the line. To enhance bandwidth it is therefore necessary to reduce l.However, in practical situations, one needs a certain value of the phase shift andthis procedure is not useful. However, by using metamaterial transmission lines,we can take advantage of certain control over the dispersion diagram of the line.This means that it is potentially possible to design a transmission line with therequired phase shift at the operating frequency, and smaller group delay than thatof a conventional line. To illustrate this, let us consider for instance the design of atransmission line with in-phase input and output signals, which can be of interestfor instance to feed an array of antennas (where in-phase signals at either radiatingelement may be required). By using a conventional transmission line, the length ofthe line must be chosen according to equation (4.24) in order to obtain a phaseshift between the input and output ports of fn ¼ 2pN, where N is an integer. Dueto the reasons already explained at the begining of this subsection, the optimum band-width is obtained when N ¼ 1, which corresponds to the shorter line. Alternatively, abalanced CRLH line operating at the transition frequency can be designed to actuallyachieve zero phase shift at that frequency. The group delay for the conventional line istC ¼ 2p/vn. Therefore, to enhance bandwidth by means of the CRLH line, it isnecessary that the group delay at the transition frequency satisfies tCRLH , tC. Inreference to LC-loaded balanced CRLH lines, the phase shift in the vicinity of thetransition frequency can be inferred by forcing vs ¼ vp in equation (3.29). This gives

bl ¼ vffiffiffiffiffiffiffiffiffiffiffiLRCR

p� 1

vffiffiffiffiffiffiffiffiffiffiffiLLCL

p : (4:25)

FIGURE 4.38 Simulated and measured frequency responses for the power divider shown inFigure 4.37. (Source: Reprinted with permission from [55].)


The first derivative with frequency gives tCRLH, and by forcing it to be smaller thantC, the following inequality results:

vn

ffiffiffiffiffiffiffiffiffiffiffiLRCR

pþ 1

vn

ffiffiffiffiffiffiffiffiffiffiffiLLCL

p , 2p: (4:26)

Two additional equations arise from the balance condition and the required character-istic impedance, Zo, of the line:

Zo ¼ffiffiffiffiffiffiLRCR

r¼

ffiffiffiffiffiffiLLCL

r: (4:27)

To satisfy inequality (4.26), and to obtain a significant bandwidth enhancement,vR (equation 3.23) must be chosen as high as possible28 and vL (equation 3.24) aslow as possible. This is consistent with the fact that to obtain a small variation ofZo in the vicinity of the operating frequency, vn ¼ vo, LR must be small (see equation3.22).

Following the previous idea, bandwidth improvement has been experimentallydemonstrated in a 1 : 4 series power divider, where zero-degree metamaterialphase-shifting lines have been used instead of 3608 lines [56]. Namely, in-phasesignals at the input of either divider branch are achieved though zero-degree designedmetamaterial transmission lines, which are placed between the different outputbranches of the divider. As compared to the conventional device, the metamaterial-based power divider exhibits an improved bandwidth. The conventional and meta-material power dividers are depicted in Figure 4.39 with their corresponding electricalcharacteristics. Another significant aspect of the metamaterial power divider is areareduction, because the metamaterial transmission lines are substantially shorterthan the conventional l lines.

Bandwidth enhancement can be achieved by using metamaterial transmissionlines in devices based on the difference of phases between certain transmissionlines. The idea is very simple. Let us consider that at the operating frequency, vn,the required phases of two transmission lines in such device are f1 and f2, respect-ively. Bandwidth depends in this case on the derivative of the phase difference withangular frequency, which is proportional to the difference in the length of such lines(expression 4.24). The larger the phase difference, the lower the bandwidth becomes.However, by replacing one of these lines with a metamaterial transmission line, it ispossible to tailor the dispersion diagram of this line to be roughly parallel to that ofthe conventional line at the frequency of interest, and hence obtain a wider band-width. This situation is illustrated in Figure 4.40.

By using the previous idea, several devices have been implemented. For instancerat race hybrid couplers have been proposed, where the phase balance for the D and Sports exhibit broader bandwidths than those obtained in the conventional implemen-tations. The topology of the conventional rat race hybrid coupler is depicted inFigure 4.41a. It is essentially a four-port device consisting of a 1.5 l ring structure

28To obtain a high value of vR, it suffices to implement the structure with a short host line.


FIGURE 4.39 Metamaterial-based (a) and conventional (b) 1 : 4 power dividers with theircorresponding measured and simulated frequency responses (only S11 (c) and S21 (d ) aredepicted). (Source: Reprinted with permission from [56]; copyright 2005, IEEE.)

FIGURE 4.40 Illustration of bandwidth enhancement by using a conventional and a left-handed transmission line. At the design frequency, vn, the phase difference, Df, betweenthe two lines is p, and this difference is roughly preserved in a wide band.


(where l is the guided wavelength at the design frequency, vn) with the ports equallyspaced in the upper half of the ring. The metamaterial counterpart of this coupler canbe implemented by substituting the 2708 (0.75 l) line section by a 2908 left-handedline designed to provide the required characteristic impedance (70.7V) and phase(2908) at the operating frequency. This was done by Okabe et al. [57] by usingthe dual-transmission-line approach, where a host line was loaded with lumped induc-tors and capacitors in shunt and series connection, respectively. The performance andsize of the device is good, with a wide bandwidth for the coupling coefficient andphase balance (see [57] for more details).

Alternatively, the metamaterial rat race hybrid coupler can be implemented bymeans of the resonant-type approach [58]. In this case, not only is the 2708 linereplaced by an artificial 2908 left-handed line, but the three 908 (right-handed) trans-mission lines are also implemented as artificial lines based on CSRRs29 (the charac-teristic impedance of these artificial lines is 70.7V). This allows us to achieve afurther controllability of the frequency dependence of phase in the lines, with theresult of excellent performance in terms of phase balance. The topology of this

FIGURE 4.41 Layout of the conventional (a) and metamaterial (b) rat race hybrid couplers.A comparative photograph of both devices can be seen in (c). The devices were fabricated onthe Rogers RO3010 substrate with dielectric constant 1r ¼ 10.2 and thickness h ¼ 635 mm.The active area (excluding access lines) of the CSRR-based hybrid coupler is 3.62 cm2,whereas the conventional one occupies an area of 10.33 cm2. (Source: Reprinted with per-mission from [58], copyright 2007, IEEE.)

29As was described in Section 4.2.3.2, these lines consist of a combination of shunt stubs and CSRRs, andthey can be described by the circuit model given in Figure 4.14.


implementation is depicted in Figure 4.41 (also included are photographs of conven-tional and metamaterial rat race hybrid couplers, designed to operate at 1.6 GHz, inorder to appreciate the achievable size reduction obtained by using metamaterialtransmission lines). The simulated and measured impedance matching, couplingand isolation for both structures are depicted in Figure 4.42. In Figure 4.43 is depictedthe phase balance for the S and D ports, namely f(S42) 2 f(S32) and f(S41) 2f(S31), respectively. The CSRR-based coupler exhibits good isolation, coupling,and matching. These magnitudes are comparable to those of the conventionaldevice. Specifically, measured power splitting between ports 3 and 4 (considering

FIGURE 4.42 Impedance matching (S11), coupling (S31, S41) and isolation (S21) for theCSRR-based hybrid coupler (a) and conventional coupler (b). The slight discrepanciesbetween simulation and measurement in the conventional coupler are attributed to fabrication--related tolerances. (Source: Reprinted with permission from [58]; copyright 2007, IEEE.)


port 1 as the input port) exhibits similar characteristics in terms of flatness. Measuredisolation and matching are comparable in both devices at the operating frequency( fo ¼ 1.6 GHz). However, phase balance in the CSRR-based rat race is clearlysuperior to that obtained in the conventional hybrid, in particular for the D input(Fig. 4.43). Namely, the phase balance variation with frequency is smaller in thenew CSRR-based hybrid and hence the operating bandwidth is enhanced. This hasbeen achieved thanks to an accurate control of the phase response of the individualartificial lines forming the proposed coupler. The design is fully compatible withplanar technology, because no lumped elements are used. With the proposeddesign, size reduction by a factor of 3 is achieved as compared to conventionalimplementation.

FIGURE 4.43 Phase balance for the D (a) and S (b) ports. (Source: Reprinted with per-mission from [58], copyright 2007, IEEE.)


The technique for bandwidth enhancement described in this section has beenapplied to other microwave components, such as phase shifters [59], where it hasbeen demonstrated in operation over one octave bandwidth.

4.3.3 Dual-Band Components

Dual-band (DB) components are devices that exhibit a certain functionality at twodifferent frequencies, f1 and f2. Such devices are of interest for modern microwaveand wireless communication systems because they make possible operation at twodifferent bands without the need to design two different mono-band (MB) circuits.Contrary to conventional (right-handed) transmission lines, which are intrinsicallyMB structures, CRLH transmission lines exhibit a DB behavior. To demonstratethat a conventional transmission line is a MB structure, we simple have to have inmind that the number of electrical parameters of a conventional line, namely theper-unit length inductance, L0 and capacitance, C0, coincide with the number ofparameters given by the specifications at a given frequency, v1, that is, the character-istic impedance, Z1, and the electrical length, b1l, where l is the physical length of thetransmission line. As is well known, the relationships between these parameters aregiven by

b1 ¼ v1

ffiffiffiffiffiffiffiffiffiL0C0

p(4:28)

and

Z1 ¼ffiffiffiffiffiL0

C0

r: (4:29)

From these equations, it is clear that we cannot obtain any arbitrary desired combi-nation of values, Z2 and b2l, at the second operating frequency, v2. Therefore, wecannot design DB components by means of conventional transmission lines.However, the situation changes if the number of electrical parameters that character-ize the line increases. This is exactly the case for the artificial CRLH transmissionlines, which are described by means of four independent parameters in LC-loadedlines or through five reactive elements in resonant-type structures. Therefore, froman analytical point of view, the previous limitation vanishes and CRLH lines areappropriate for the design and synthesis of DB components. Essentially, to designa DB component means tailoring the dispersion diagram and/or the characteristicimpedance in order to achieve the required values of impedance and phase at thedesired operating frequencies, and this can be done by means of CRLH transmissionlines. To clarify this, we will give an example. The idea is to achieve by means of asingle CRLH line cell described by the model of Figure 3.3, a DB l/4 impedanceinverter (with normalized characteristic impedance K ¼ 1), operative at frequenciesf1 ¼ 1 GHz and f2 ¼ 2 GHz. To achieve this, we can take benefit of the left-handedand right-handed behavior of the CRLH cell. To simplify the calculation, we will con-sider the balanced case. The idea behind this design is to set the transition frequency,vo ¼ vs ¼ vp, between the operating frequencies, v1 and v2, and determine the line


parameters so that at v1 and v2 the electrical length of the line is b1l ¼ 2p/2 andb2l ¼ þp/2, respectively, with the characteristic impedance being Zo ¼ 50V atboth frequencies. To this end, equations (3.21) and (3.22) are solved subject to thebalanced case and to the previous specifications. The resulting electrical parametersof the line are CL ¼ 0.796 pF, LL ¼ 3.98 nH, CR ¼ 3.18 pF, and LR ¼ 15.92 nH.In Figure 4.44 are depicted the dispersion relation of the structure, and the depen-dence of the characteristic impedance on frequency. The required values for bothbl and Zo have been obtained at the operating frequencies. As the sign of thephase shift is irrelevant for impedance inversion, this is clearly a good example ofthe DB behavior of CRLH transmission lines. Other combinations of phase shifts(all being a positive or negative multiple of p/2) at the desired frequencies arealso possible. Depending on the required phase variation, more than one cell mightbe necessary.

Dual-band prototype components based on the CRLH transmission line concepthave been developed by Lin et al. [60]. To illustrate the possibilities of the approach,Figure 4.45 includes a DB branch line coupler, and its measured frequency response.At the operating frequencies, the required impedance inverters have been designed to

FIGURE 4.44 Dispersion diagram (a) and characteristic impedance (b) for the CRLH modeldesigned to achieve dual-band operation according to the frequencies and requirementsexplained in the text.


exhibit phase shifts of þ908 and þ2708, as indicated in the schematic. The circuit hasbeen implemented by cascading purely right-handed (PRH) lines (implementedthrough conventional microstrip sections) and purely left-handed (PLH) lines (rea-lized by means of SMT chip components) for either branch of the coupler.Operation at the design frequencies ( f1 ¼ 930MHz and f2 ¼ 1780MHz) is clearlyachieved, with near 3 dB transmission between the input port and both the coupledand through ports.

4.3.4 Coupled-Line Couplers

Metamaterial transmission lines can be also applied to the optimisation of coupled-line couplers. These are four-port devices consisting of a pair of coupled lines, asindicated in Figure 4.46, with diverging access lines for each port. Although topolo-gically identical, there are two types of coupled-line couplers, namely the backwardcoupler and the forward coupler (Fig. 4.46). The principle of operation of suchcouplers has been exhaustively described elsewhere [61], but it is concisely

FIGURE 4.45 DB branch line coupler (a), measured frequency response (b) and schematicsof the coupler for the two operating frequencies (c) f1 and (d ) f2. (Source: Reprinted withpermission from [60]; copyright 2004, IEEE.)


reproduced here to understand the main limitations of these type of couplers and theadvantages of metamaterial-based implementations of them.

In the backward wave coupler, the coupled port is that port sharing the same refer-ence plane as the input port, whereas the isolated port is the crossing port (in referenceto the input port). Under the assumption of a symmetric structure, even and odd modeanalysis lead us to the coefficients of the S-matrix. The condition to obtain perfectmatching at the input port is given by

Zo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiZoeZoo

p(4:30)

where Zoe and Zoo are the characteristic impedance for the even and odd modes,respectively, and Zo is the reference impedance of the port. Under these conditions,the following results are obtained:

S11 ¼ 0, (4:31a)

S21 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

pcosfþ j sinf

, (4:31b)

S31 ¼jk sinfffiffiffiffiffiffiffiffiffiffiffiffiffi

1� k2p

cosfþ j sinf, (4:31c)

and

S41 ¼ 0, (4:31d)

where it is assumed that the phase shift f (or the phase velocity) for the even and oddmodes is roughly the same (which is in turn an assumption valid for quasi-TEM wavepropagation). Coupling (i.e., S31) depends on the electrical length of the coupled linesand on the coupling factor, k, given by

k ¼ Zoe � ZooZoe þ Zoo

: (4:32)

FIGURE 4.46 Topology of the backward (a) and forward (b) wave couplers with indicationof port designations.


The optimum coupling is obtained for f ¼ p/2. In this case, coupling is C ¼ jS31j ¼k. Thus, in these backward wave couplers, maximum coupling depends on the differ-ence between the characteristic impedance for the even and the odd modes. In prac-tice, to achieve significant coupling levels, a high contrast between these impedancesis required, and this is difficult to achieve in edge-coupled couplers, because too smalldistances between the coupled lines are required for this purpose. In other words, thecoupling levels achievable with these couplers are very limited. This is the maindrawback of these type of couplers. However, they are relatively small, because908 lines (or shorter, depending on the required coupling) are used, and bandwidthis reasonably good.

Let us now briefly consider the forward wave couplers. In these devices, the dis-tance between the coupled lines is deliberately large (i.e., Zoe � Zoo and k � 0). Thismeans that signal is not transmitted to the so-called coupled port of the backwardwave coupler (which is now the isolated port). Indeed, according to equations(4.31) and (4.32), all the input power should be collected in the through port. Thistrivial situation may change, however, if the phase velocities for the even and theodd modes are not identical, which is what actually occurs in real structures, thatis, exhibiting quasi-TEM, rather than TEM, wave propagation. In this case, the scat-tering parameters are

S11 ¼ 0, (4:33a)

S21 ¼ exp�j(be þ bo)l

2

� �cos

(be � bo)l2

� �, (4:33b)

S31 ¼ 0, (4:33c)

and

S41 ¼ �j exp�j(be þ bo)l

2

� �sin

(be � bo)l2

� �, (4:33d)

where be and bo are the phase constants for the even and odd modes, respectively,and l is the length of the coupled lines. The coupled port is the crossing port andcoupling is now given by C ¼ jS41j. Hence, forward coupling is obtained, which jus-tifies the name given to the device. According to equation (4.33d), coupling dependson the length of the device, and optimum coupling is obtained for the coherencelength, lc, given by

lc ¼p

be � boj j : (4:34)

Due to the comparable phase velocities for the even and the odd modes, the coher-ence length (or the typical length to achieve substantial couplings) is too long for


practical applications, this being the main limitation of this type of coupler. In thecase of asymmetric forward wave couplers, the analysis is more complicated, butthe same conclusion concerning device length is inferred (expression 4.34 holdsbut with the phase constants modified following be ! bc and bo ! bp to accountfor the c and p modes of the asymmetric structure [61,62]).

Let us now analyze how the previous limitative aspects can be improved throughthe use of metamaterial transmission lines. With regard to forward couplers, the ideais to replace the coupled lines (or eventually only one of the coupled lines) by left-handed lines [63,64]. If both lines are backward lines, power is coupled forward(i.e., between port 1 and 4 in Fig. 4.46), although due to the left-handed nature ofthe coupled lines, wave propagation is backward. If only one of the coupled linesis backward, and it corresponds to that line connecting the input and through ports(line 1–2 according to Fig. 4.46), injected power into port 1 propagates towardsport 2. However, due to left-handedness, waves propagate backward towards port 1.As coupling in this kind of couplers preserves the phase direction at both lines (coup-ling occurs through evanescent waves), it follows that phase and power in line 3–4propagates to port 3, which is now the coupled port. Thus, power is coupled back-ward, rather than forward. This is a fundamental difference with the conventional(symmetric or asymmetric) forward wave coupler and with the forward couplerimplemented by means of two coupled left-handed lines.30 However, the couplingmechanism is identical. The key advantage of using left-handed lines in these cou-plers is the possibility of designing devices of small size and with enhanced coupling(compared to conventional forward wave couplers). The reason is the unusual dis-persion diagram of left-handed lines. For couplers with identical left-handedcoupled lines (symmetric coupler), an analysis similar to that shown above for con-ventional forward couplers leads us to the conclusion that coupling increases bydecreasing frequency31 (in contrast to conventional couplers, where high frequenciesor long coupled lines are required to achieve maximum coupling). In asymmetric cou-plers implemented by a conventional and a left-handed line, the phase constants ofexpression (4.34) can be replaced with the phase constants of the conventional andleft-handed lines [64].32 As the phase constant of the left-handed line is negative,the sign in the denominator is positive and the coherence length is thus reduced.Alternatively, size reduction and high coupling in these type of couplers have beeninterpreted on the basis of coupled-mode theory [66]. To this end, the coupled-mode dispersion diagram for the coupled-line coupler has been inferred, and it hasbeen concluded that operation in the stop-band for the coupled-mode system leadsto continuous power leakage from one line to the other [66].

As an example, Figure 4.47 illustrates a fabricated prototype device correspondingto a mixed conventional/left-handed asymmetric coupler, where the left-handed line

30To avoid the incoherence of calling forward couplers to those metamaterial-based couplers implementedby means of only one left-handed line (where power is actually coupled backwards), Caloz and Itoh havetermed these devices phase couplers in their textbook [65].31This is due to the frequency dependence of the phase constant in left-handed lines.32This is valid for very weak coupling between the lines.


is implemented by LC loading a microstrip line. Near 0 dB backward coupling in abroad fractional bandwidth of roughly 50% is obtained in this coupler [64] (seeFig. 4.47). Alternatively, these mixed couplers have been implemented by loadingthe left-handed line with CSRRs and series gaps (Fig. 4.48) [67]. Simulation of elec-tric field propagation (Fig. 4.49) indicates that power is coupled backward, asexpected. The measured S-parameters for the coupler of Figure 4.48 are depictedin Figure 4.50, where they are compared with the simulated S-parameters of a conven-tional (backward) 908 parallel coupled-line coupler with identical distance betweenthe coupled lines and substrate. For the metamaterial-based coupler, the couplinglevel at the design frequency, 4.3 GHz, is in the vicinity of 24 dB, whereas it islower than 220 dB in the conventional backward coupler. It is also possible todesign DB mixed conventional/left-handed couplers by loading the left-handedline of the coupler with two sets of CSRRs of different dimensions, as Figure 4.51illustrates [67].

With regard to the improvement of the conventional backward wave coupler pre-sented above, this is relatively complex and it has been considered in detail in [65,68].The considered coupler, composed of two identical left-handed balanced lines, issymmetric and even/odd analysis is thus applicable in this case. Operation of thecoupler in the stopband for either mode (even/odd) leads to enhanced bandwidth

FIGURE 4.47 Mixed conventional/left-handed coupled-line coupler (a), simulated (b), andmeasured frequency responses (c). (Source: Reprinted with permission from [64]; copyright2004, IEEE.)


FIGURE 4.48 Top (a) and bottom (b) photographs of the fabricated CSRR-based coupler.The device consists of a conventional line and a three-cell CSRR/gap loaded left-handed line.To further appreciate the details of the topology, the layout of the three-cell left-handed lines isdepicted in (c). The device has been fabricated in the Rogers RO3010 substrate (dielectric con-stant 1r ¼ 10.2, thickness, h ¼ 1.27 mm). CSRR dimensions are external radius, rext ¼ 2 mm,width, c ¼ 0.2 mm and separation, d ¼ 0.2 mm. Gap distance is 0.2 mm and the period of thestructure is 4.2 mm. (Source: Reprinted with permission from [67]; copyright 2006, JohnWiley& Sons.)

FIGURE 4.49 Simulation of the electric field propagation in a 15-cell mixed conventional/left-handed coupler based on CSRRs. CSRR dimensions and separation are identical to thosevalues of Figure 4.48.


and high coupling. As pointed out in [65], any arbitrary level of coupling can beachieved with these couplers, even with a large distance between the coupledlines. The detailed analysis of this type of couplers is outside the scope of thisbook, but the authors recommend reference [65] to the interested reader.

4.4 ANTENNA APPLICATIONS

Antennas have become one of the most exciting applications of metamaterials dueto the possibility of significantly improving their performance. Indeed, many

FIGURE 4.50 Measured S-parameters of the CSRR-based coupler of Figure 4.48 (a) andsimulated S-parameters for a 908 parallel coupled-line coupler with similar dimensions (b).(Source: Reprinted with permission from [67]; copyright 2006, John Wiley & Sons.)


papers have been published since the boom of metamaterials in 2000. We willstart by describing some pioneering work that focuses on the elementalproperties of antennas and, later, we will continue with antenna applications ofmetamaterials.

It has been well established that pairs of media with positive and negative refrac-tion indexes are analogous to reactive impedances of opposite sign, in that they areable to resonate. Alternatively, similar responses can be achieved by means ofpairs of media exhibiting positive or negative permittivity or permeability [69].Therefore, it follows that metamaterial structures can be useful to introducechanges in antenna impedance and, consequently, to obtain an improvement inantenna response. This was proposed in [70], where the authors show that it ispossible to match an electrically small dipole to the free-space impedance thanksto the use of a metamaterial spherical cover with negative index of refraction.Further details concerning the expressions of the radiated fields, radiated power,lumped-element equivalent-circuit models, quality factor, and other properties ofsuch elemental antennas when enclosed in right-handed or left-handed media, arepresented in [70]. An interesting tunnel-effect interpretation of this impedance-matching phenomenon and a reciprocal scattering description are presented in [69].Another surprising consequence of the structure proposed in [70] is that an importantincrease in the radiated power and a reduction of the reactance are simultaneouslypossible, thus allowing circumvention of the Q radiation limit of Chu [70].

FIGURE 4.51 Topology of the dual CSRR-based coupler (a) and simulated S-parameters(b). (Source: Reprinted with permission from [67]; copyright 2006, John Wiley & Sons.)

4.4 ANTENNA APPLICATIONS 253

A paper dealing with the study of the radiation properties of a traveling wave at theinterface between a left-handed and a right-handed medium was published byAlu and Engheta [71], where the electromagnetic field distribution and the wavevec-tor and Poynting vector orientation in both left-handed and right-handed half-spaces were calculated. Further analysis of metamaterial radiating structures can befound in [72].

The increasing effect on radiated power has encouraged research in antenna appli-cations of metamaterials. For example, in [73] it is proposed to use an annular left-handed material ring inside a microstrip patch resonator antenna, where the resonancefrequency can be decreased by proper design of ring dimensions relative to the rest ofthe patch antenna. This suggests, at least in theory, the possibility of very compactantenna design for relatively low frequencies. A theoretical study of patch antenna,including a left-handed medium, is presented in [74], and the use of SRRs as amagneto-dielectric substrate for patch antennas is proposed in [75].

The first experimental results of radiation in left-handed media were motivated bythe demonstration of reversed Cerenkov radiation as predicted by Veselago [1]. Inthis regard, two simultaneous contributions appeared in 2002 [76,77], where back-ward, leaky wave radiation in dual transmission lines, analogous to reversedCerenkov radiation, was experimentally demonstrated. Indeed, backfire-to-endfireradiation in these structures can be achieved by properly engineering the dispersiondiagram.33 A very interesting property of these leaky-wave antennas is their operationin the fundamental mode.34 This reduces the feeding structure complexity and allowsfor broadside radiation, thanks to the nonzero group velocity (traveling wave).Moreover, continuous backfire-to-endfire radiation can be electronically achieved,which is very interesting for the design of electronically scanned antennas [78–83](see Fig. 4.52). Backward leaky waves can also be achieved in SRR-based left-handed structures [84].

Through leaky wave analysis, the study of the radiation characteristics of left-handed (or CRLH) lines (including resonant and nonresonant structures) is possible.Indeed this analysis can also be extended to other periodic structures based on meta-materials not necessarily providing backward leaky waves. Comparison between thephase constant, b, and the free-space wavenumber, ko, can be used to predict the fre-quency bands where these structures can radiate. Specifically, if jbj , ko, then radi-ation may occur; and the angle, u, between the radiation beam and the propagationaxis can be calculated as

cos u ¼ b

ko, (4:35)

where this angle can be positive or negative, depending on the sign of the phase con-stant. According to this, backward leaky wave radiation is expected if jbj , ko and

33Infact, the dual transmission lines exhibit a CRLH behavior that makes it possible to achieve backward orforward leaky waves.34Conversely, operation of conventional periodic-type leaky-wave structures is based on spatial harmonics.


b, 0, whereas forward leaky wave radiation will occur if jbj , ko but b . 0. It hasbeen demonstrated that CPW structures simply loaded with SRRs are able to produceforward leaky waves (Fig. 4.53) [84]. Conversely, if shunt-connected strips are addedto the structure, a left-handed pass band results above the resonance frequency of theSRRs, and the radiation is backward (Fig. 4.54) [84]. The dispersion characteristics ofthese structures are depicted in Figure 4.55. In Figure 4.55a, the existence of a left-handed band for the structure shown in Figure 4.54 is verified. It corresponds to thezone of the dispersion diagram where the phase constant and its derivative withfrequency have opposite signs (antiparallel group and phase velocities). The attenu-ation constant is small in most of this region. The upper part of such a band lies withinthe radiation region (jbj, ko), hence predicting the existence of a frequency rangewhere the Bloch mode becomes leaky. As has been explained, the left-handed

FIGURE 4.52 Prototype of electronically scanned CRLH microstrip leaky-wave antenna (a)and measured radiation patterns at different bias voltages (b). (Source: Reprinted with per-mission from [83]; copyright 2004, IEEE.)


behavior of the structure will result in a reversal of the direction predicted for theleaky wave radiation. The phase-matching condition along the interface betweenthe left-handed structure and the medium where the energy is radiated (free space,and therefore a right-handed medium) implies that the radiated energy will necess-arily have to propagate in the opposite direction. This can also be explained bymeans of equation (4.35), as b is negative for waves propagating energy from leftto right (which is the usual convention), and therefore u will be included in thesecond quadrant, that is, corresponding to radiation from broadside (u ¼ 908) to back-fire (u ¼ 1808) direction. This phenomenon can be seen as analogous to the reversalof Cerenkov radiation in left-handed materials, as predicted by Veselago in hisseminal work [1].

FIGURE 4.53 Layout of a forward leaky-wave structure consisting of a CPW loadedwith SRRs in the back substrate side (a) and measured far-field radiation pattern (b).E-plane (continuous line) and H-plane (dashed line). (Source: Reprinted with permissionfrom [84].)


FIGURE 4.54 Layout of a backward leaky-wave structure consisting of a CPW loaded withSRRs in the back substrate side and shunt-connected strips in the upper side (a) and measuredfar-field radiation pattern (b). E-plane (continuous line) and H-plane (dashed line). In (c) a CSTMicrowave Studio simulation of the E-plane pattern as frequency changes is depicted. (Source:Reprinted with permission from [84].)


Figure 4.55b shows the dispersion diagram for the CPW structure loaded onlywith SRRs (Fig. 4.53). In this case, there is a frequency band where strong rejectionoccurs due to the negative value that magnetic permeability exhibits in that band. Inthat region the attenuation constant is much larger than the phase constant. Both therejection band and the lower part of the second pass band are included in the radiationregion (jbj , ko). However, the high attenuation constant present in the stop-bandprevents the structure from radiating, making the leakage almost completely reactiveat these frequencies [84]. On the other hand, in the low-frequency region of thesecond pass band, the structure radiates (small attenuation constant), and because itis right-handed (parallel phase and group velocities), the radiation beam will be con-tained in the first quadrant, from broadside to endfire direction (positive value for b).

Several envisaged applications of metamaterial-based leaky-wave structuresinclude tunable antennas using distributed thin-film ferro-electric and ferro/i-magnetic materials, anisotropic minimum-impedance-path metamaterials forbeam-forming structures, full-space scanning 2D leaky-wave antennas for a possiblyinexpensive alternative to complex conventional phased arrays, compact-range wavegeneration and measurements, or endfire antennas. More details of these possibleapplications are described in [79,80].

PROBLEMS

4.1. Synthesis of filters based on the hybrid model. Under the approximation thatthe series reactance of the circuit model of Figure 4.18b is constant along theleft-handed passband, demonstrate that the shunt impedance is Zp ¼ jZo/2 andinfinity, respectively, at the 23 dB frequencies (v1 and v2) of the structure.From this, infer expressions (4.9) and (4.10).

4.2. Tunable notch filters based on VLSRRs. In reference to Figure 4.33,explain the variations in the frequency response (gap width and rejectionlevel) that result under different bias polarizations. Hint: Express the simplifiedcircuit model of Figure 4.34 as that depicted in Figure 3.23b (by taking intoaccount the parallel resistance of the resonant tank) and obtain the Q-factor ofthis tank.

4.3. Synthesis of left-handed lines based on CSRRs. As was discussed in Chapter3, the left-handed transmission lines based on the dual and resonant-typeapproach can be considered as artificial transmission lines, where impedanceand phase can be tailored over a wide margin at a given operating frequencywithin the band. Specifically, for the circuit depicted in Figure 3.38 (whichmodels a CSRR-based left-handed line), the phase varies in the interval(2p, 0) in the backward wave band, whereas the characteristic impedance iszero at the onset of the left-handed band, then increases progressively, and(ideally) tends to infinity at the upper edge of the band. Demonstrate that itoccurs if the resonance frequency of the series branch is higher than the reson-ance frequency of the CSRRs (i.e., the series capacitance dominates over the


series inductance in the backward wave band). Under these conditions (i.e.,neglect the series inductance L), obtain the equations that univocally determinethe circuit parameters Cg, C, Lc, and Cc from the required impedance, Zn, andphase fn at the operating frequency fn. The two additional inputs to solve thisproblem are the limits of the backward wave band, fL and fH.

4.4. Bandwidth enhancement in metamaterial transmission line based devices.It was demonstrated in Section 4.3.2 that metamaterial transmission lines canbe used for the design of enhanced-bandwidth components, and several proto-type device examples are provided in that section. One of the devices is a1 : 4 power divider, where the 3608 transmission lines that are used in the

FIGURE 4.55 Phase (bold line) and attenuation constant (thin line) for the structures ofFigure 4.54 (a) and 4.53 (b). The radiation region is shown with dotted lines. (Source:Reproduced with permission from [84].)

PROBLEMS 259

conventional implementation to achieve in-phase signals, are substituted byzero-degree artificial lines in the metamaterial counterpart. Discuss if it isalways possible to improve the bandwidth of conventional lines through meta-material transmission lines regardless of the electrical length of such lines.

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CHAPTER FIVE

Advanced and Related Topics

5.1 INTRODUCTION

In previous chapters we have developed the central concepts of metamaterials scienceand its main technological applications as they are perceived by the authors. Inaddition, metamaterial concepts often act as shortcuts to many other concepts andapplications that probably would not be developed (or would be developed in a differ-ent way) without the current interest in metamaterials. Some of these topics aredescribed in this chapter. The choice is, of course, arbitrary, and has much to dowith author’s personal experience and taste. We have chosen those topics that, inour opinion, would likely give rise to practical applications in the near future. Wehope that our intuition will not be too wrong.

The chapter begins with the analysis of admittance surfaces designed from SRRsand CSRRs (see Chapters 2 and 3), whose reflection and transmission properties canbe related through Babinet principle. We hope that these structures may find appli-cation as frequency-selective surfaces, polarizers, and polarization converters. Wealso feel that the reported admittance surface approach could be useful for the analysisof two-dimensional micro- and nanostructured metal–dielectric surfaces with exoticelectromagnetic properties (meta-surfaces).

Magneto-inductive waves—and the complementary electro-inductive waves—excited in one-, two-, and three-dimensional arrays of metamaterial elements arethe next topic analyzed in this chapter. Magneto-inductive waves are of theoreticalinterest as a specific kind of waves that can be excited in metamaterial structures.They could find application in alternative waveguiding devices and systems,which may exploit their slow-wave characteristics. They could be also useful fornear-field subdiffraction imaging.

Imaging is the last topic addressed in this chapter. Subdiffraction imaging is oneof the most exciting applications of metamaterials. In this chapter, we will analyze


267

some subdiffraction imaging devices—other than the left-handed slab analyzed inChapter 1—that provide similar results with a simpler design. These devicesinclude plasmonic slabs, ferrite slabs, and coupled arrays of magnetic resonators.Finally, this last section also includes a brief report on some newly proposedimaging devices, based on light canalization.

5.2 SRR- AND CSRR-BASED ADMITTANCE SURFACES

Both SRRs and CSRRs are planar small resonators, which can be easily printed on alow-loss dielectric circuit board in order to configure resonant surface arrays withinteresting electromagnetic properties. Duality [1] and the Babinet principle [2]impose that the behavior of SRR and CSRR surface arrays must be approximatelydual (small deviations from this duality may arise from the effects of the dielectricsubstrate on which SRRs and CSRRs are printed). Moreover, as far as the electricalsize of the elements and the periodicity can be considered small, the surface admit-tance concept appears a useful tool for the analysis of such configurations.Throughout this section, these concepts will be developed in order to provide afirst approach to the behavior and applications of SRR and CSRR surface arrays.

5.2.1 Babinet Principle for a Single Split Ring Resonator

In the most general case, when it is illuminated by some external field E0, B0, a singleedge-coupled SRR (EC-SRR) will show a set of polarizabilities given by (seeChapter 2)

mz ¼ ammzz B0

z � aemyz E

0y , (5:1)

py ¼ aeeyyE

0y þ aem

yz B0z , (5:2)

and

px ¼ aeexxE

0x , (5:3)

where, near the first EC-SRR resonance, the polarizabilities in expressions (5.1) and(5.2) take the general form

a ¼ a0v20

v2� 1

� ��1

, (5:4)

where a0 is a geometrical factor (the remaining polarizability aeexx has a nonresonant

behavior at v ¼ v0). The effect of an incident electromagnetic field on a singleEC-SRR is illustrated in Figure 5.1a. The magnetic and electric dipolar moments(5.1)–(5.3) are generated in the SRR, and, in the long wavelength approximation,determine the scattered fields E0 and B0. The total fields are the superposition ofthe incident and the scattered fields. Let us now consider the complementary

268 ADVANCED AND RELATED TOPICS

screen, that is, the complementary resonator (CSRR). If the screen is a perfect con-ductor of negligible thickness, and the effects of the dielectric substrate are ignored(or it has a negligible dielectric susceptibility), the behavior of this CSRR can bededuced from classical diffraction theory and the Babinet principle [2]. The CSRRbehavior when it is illuminated by the complementary fields

E0c ¼ cB0; B0

c ¼ �(1=c)E0 (5:5)

(c is the velocity of light in vacuum), incident from the left (z, 0), is illustrated inFigure 5.1b. At the right-hand side of the screen (z. 0), the fields are those producedby the electric and magnetic dipolesm and p. These electric and magnetic dipoles aregiven by

pz ¼ beezzE

0z � bem

yz B0y , (5:6)

my ¼ bmmyy B0

y þ bemyz E

0z , (5:7)

and

mx ¼ bmmxx B0

x , (5:8)

where, from the Babinet principle [2] and the well-known expressions for the electro-magnetic fields of the electric and magnetic dipoles, it follows that [4]

bmmxx ¼ �c2aee

xx; bmmyy ¼ �c2aee

yy; bemyz ¼ �aem

yz ; beezz ¼ � 1

c2ammzz : (5:9)

According to diffraction theory [2], the fields at the left-hand side of the screen (z, 0)are the superposition of the incident field (5.5), the field that would be reflected by aperfect metallic screen at z ¼ 0 (E0,r

c , B0,rc ), and the field created by some

magnetic and electric dipoles, which are the opposite of equations (5.6)–(5.8).Note that this change of sign for the induced dipoles at the right- and left-handsides of the screen ensures that both the total magnetic polarization perpendicularto the screen and the total electric polarization parallel to the screen vanish, asmust be for a plane screen.

FIGURE 5.1 Illustration of the behavior of an SRR (a) and a CSRR (b) when they are illu-minated by an external field E0; B0 (a) or E0

c ¼ cB0; B0c ¼ �(1=c)E0 (b) coming from z, 0.

5.2 SRR- AND CSRR-BASED ADMITTANCE SURFACES 269

5.2.2 Surface Admittance Approach for SRR Planar Arrays

Let us now consider the behavior of a plane array of EC-SRRs. For simplicity, asquare array of periodicity awill be considered (see Fig. 5.2). The considered incidentfield will be a plane wave of arbitrary polarization, incident on this array from the left-hand side (z , 0). As far as the size of the SRRs can be considered small with regardto the wavelength, the far field produced by the surface currents on the array can beapproximated by the field created by the average currents on it. Such average currentsarise from the mean surface electric Ps and magneticMs polarizations generated in theSRR array. Such polarizations are generated from the electric and magnetic fields

FIGURE 5.2 A planar square array of EC-SRRs (a), and its complementary screen, a planararray of CSRRs (b). In the calculations both arrays are supposed to extend to infinity at bothsides.


incident upon each SRR through expressions (5.1)–(5.3). Therefore, there is a linearrelation between the mean surface currents and the average fields on the array of thekind [5]

Js,xJs,y

� �¼ Yxx Yxy

Yyx Yyy

� �� Ex

Ey

� �: (5:10)

The surface current density Js comes from the surface polarization Ps ¼ pya�2 y, andthe surface magnetization Ms ¼ mza�2 z through [6]

Js ¼ jvPs � z�rMs,z, (5:11)

where the last term comes from the identity r�Ms ; � z�rMs,z. The surfaceadmittances Yi, j; i, j ; x, y, can be deduced from the SRR polarizations (5.6)–(5.8)through the appropriate homogenization procedure.

The more simple homogenization procedure equates the local field seen by eachSRR with the average field in equation (5.10).1 In such a approximation—which isstrictly valid only for very sparse arrays—the dipolar moments px, py, and mz aregiven by expressions (5.1)–(5.3) with E0 ¼ E and B0 ¼ B. Taking into accountFaraday’s law (r� E ¼ �jvB) and equation (5.11), the averaged surface currentdensity on the SRR array can be expressed as

Js,x ¼ jvaeexx

a2Ex �

aemyz

a2@Ey

@yþ j

ammzz

va2@2Ey

@x@y� @2Ex

@y2

� �, (5:12)

and

Js,y ¼ jvaeeyy

a2Ey þ

aemyz

a2@Ex

@y� j

ammzz

va2@2Ey

@x2� @2Ex

@x@y

� �, (5:13)

which, for the particular case of an incident plane wave (E0 / exp( jvt � jkr))reduces to [5]

Yxx ¼ jvaeexx

a2þk2ya

mmzz

v2a2

( ), (5:14)

Yyy ¼ jvaeeyy

a2þk2xa

mmzz

v2a2

� �, (5:15)

Yxy ¼ jvaemyz kyva2

� kxkyammzz

v2a2

� �, (5:16)

1An example of another—and more accurate—homogenization procedure is developed in the next sectionfor the analysis of magneto-inductive surfaces. In this section, however, we will use this simpler homogen-ization, because we are mainly interested in a qualitative description of the most important physical effectsassociated with this kind of meta-surface.


and

Yyx ¼ jv �aemyz kyva2

�kxkyamm

zz

v2a2

� �, (5:17)

where kx and ky are the transverse components of the wave vectors of the incidentfield. It is worth noting that, for loss-less SRRs, aem

yz is imaginary, whereas aeeyy and

ammzz are real numbers (see Chapter 2). Therefore Yyx ¼ �Y�

x,y, as is imposed byenergy conservation.

It should be kept in mind that expressions (5.14)–(5.17) were obtained from thesimplest approximation, only valid for sparse arrays of weakly coupled SRRs.However, (equation 5.10) is completely general, provided the long wavelengthapproximation is assumed. From this expression, provided the surface admittancematrix is known, the reflection and transmission coefficients of the array can beobtained (see Sections 5.2.4 and 5.2.5) for any angle of incidence.

5.2.3 Babinet Principle for CSRR Planar Arrays

Let us now consider the complementary array of Figure 5.2b. It is possible, in prin-ciple, to develop an admittance approach for this planar array, similar to that devel-oped in the previous paragraphs. However, if the effects of the screen thickness andconductivity, as well as the dielectric board polarizability, are neglected, the trans-mission and reflection coefficients of such as array can be obtained directly fromthose of the SRR array by applying the Babinet principle.

Let us consider the complementary incident field (5.5) on the CSRR array. TheBabinet principle [2] states that the total magnetic field of the SRR array, B, andthe total electric field of the CSRR array, E c, at the shadowed region (z . 0) arerelated by

cBþ Ec ¼ cB0: (5:18)

Therefore, the transmission coefficient for the CSRR array illuminated by thecomplementary wave (5.5), tc, is related to the transmission coefficient for the SRRarray, t, by

t þ tc ¼ 1: (5:19)

Therefore,

tc ¼ �r and rc ¼ �t, (5:20)

where r and rc are the reflection coefficients for the SRR array and for the CSRR array(illuminated by the complementary wave), respectively. Therefore, as is expected, thetransmission and reflection coefficients for the SRR and the CSRR array interchangetheir roles when the CSRR array is illuminated by the complementary incident wave.

The aforementioned results are valid for the copolarized components of thereflected and transmitted waves (i.e., for the reflected and transmitted waves having


the same polarization as the incident one). However, because the admittance matrix inexperssion (5.10) may not be diagonal, cross-polarized reflected and transmittedwaves can be generated at the SRR and CSRR arrays. If the cross-polarization trans-mission coefficients, t0 and t 0c, are considered, from expression (5.18) it follows thatt0 þ t0c ¼ 0. And, as it must be that r0 þ t0 ¼ 0 for the cross-polarization coefficients,we finally obtain

t0 ¼ �r0 ¼ �t0c ¼ r0c: (5:21)

As has already been mentioned at the beginning of this section, the above results arestrictly valid for perfectly conducting screens of negligible thickness, and for sub-strates of negligible polarizability. In practice, these results must be considered as afirst approximation. For realistic structures, the most important deviations from theabove results come from the ohmic losses in the screens, and from the deviationsin the frequency of resonance between the SRR and CSRR arrays, due to theeffect of the substrate. This last effect can be approximately taken into account byintroducing the CSRR frequency of resonance in place of v0 in equations (5.1)–(5.3).

5.2.4 Behavior at Normal Incidence

For normal incidence kx ¼ ky ¼ 0 and the admittance matrix in equation (5.10) isdiagonal. Therefore, no cross-polarization effects appear. That is, for the two orthog-onal polarizations (E0 ¼ E0 x, and E0 ¼ E0 y), the transmitted and reflected waveshave the same polarization as the incident wave. As aee

yy is resonant and aeexx is not, the

behavior of the SRR array for both orthogonal polarizations is quite different. Forwaves polarized with the electric field along the x-axis, the SRR array is almost trans-parent. Of more interest is the behavior for waves polarized along the y-axis. A strongreflectivity can be expected for such waves near the SRR frequency of resonance.Therefore, near such frequency, the SRR array acts as a polarizer, which is transparentfor x-polarization and opaque for y-polarization.

Let us now analyze in detail the behavior for the most interesting polarization ofthe incident wave: E0 ¼ E0 y exp( jvt � kz). In such a case, from equations (5.10)and (5.14)–(5.17), it follows that the SRR array is characterized by a surfaceadmittance Ys given by

Js ¼ YsE; Ys ¼ jvaeeyy

a2: (5:22)

The transmission and reflection coefficients for the SRR array can now be obtainedfrom equation (5.22) and the appropriate boundary conditions for the tangential com-ponents of the electric and magnetic fields at the screen. The final result is [5]

t ¼ 1þ r ¼ 2Ysh0 þ 2

, (5:23)


where h0 ¼ffiffiffiffiffiffiffiffiffiffiffiffim0=10

pis the free-space impedance. This equation predicts a total

reflection peak when Ys ! 1, that is, at the frequency of resonance of the SRRs.It also predicts a total transmission frequency point when Ysh0 ¼ 0. According tothe expression for aee

yy given in Chapter 1, this point must be located above thefrequency of resonance of the SRRs.

The behavior of the complementary CSRR array for complementary excitation,that is, for a plane incident wave polarized along the x-axis, can be deduced fromthe above equations and from equation (5.20): a strong transmission peak is predictedat the CSRR frequency of resonance, as well as a total reflection peak just above thisfrequency. These results are exact only for the very strict conditions mentioned at theend of Section 5.2.3.2 However, they still describe most of the physics of the analyzeddevices. In practice, deviations from Babinet correspondence (5.20) can be expectedas a consequence of substrate effects and ohmic losses. In addition, deviations of theaforementioned SRR-reflection and CSRR-transmission peaks frequencies from theresonant frequencies of the SRR and CSRR must be expected as a consequence ofthe coupling between array elements (which were ignored in equations 5.14–5.17).

An experimental confirmation of the behavior described in the previous paragraphis shown in Figure 5.3, where the aforementioned strong SRR-reflection and CSRR-transmission peaks can be clearly appreciated. As was already mentioned, the fre-quency shift between these peaks can be attributed to the presence of the dielectricsubstrate, and the deviation from total transmission in the CSRR array to ohmiclosses. It may be worth mentioning here that the presence of such extraordinarytransmission/reflection peaks at normal incidence is a consequence of the cross-polarization effects in the EC-SRR, already analyzed in Chapter 2. In fact, theseeffects would dissapear if nonbianisotropic elements—such as the NB-SRRs orBC-SRRs reported in Chapter 2—were used. Finally, it may be worth noting (inorder to judge the accuracy of the proposed models) that the frequency of resonanceof the SRRs considered in Figure 5.3, predicted by the model reported in Chapter 2, isv0 ¼ 2p� 4:41 GHz, and the frequency of resonance of the CSRRs, predicted bythe model reported in Chapter 3, is v0 ¼ 2p � 4.65 GHz.

5.2.5 Behavior at General Incidence

In order to obtain the transmission and reflection coefficients for an arbitrary angle ofincidence, the boundary conditions on the surface z ¼ 0,

Js,x ¼ Hiy þ Hr

y � Hty, (5:24)

Js,y ¼ �Hix � Hr

x þ Htx, (5:25)

0 ¼ Eix þ Er

x � Etx, (5:26)

2And, because the particular homogenization procedure has a limited scope, only for weakly coupledSRRs/CSRRs arrays.


and

0 ¼ Eiy þ Er

y � Ety, (5:27)

must be substituted in equation (5.10). Using Faraday’s law, expressions (5.24) and(5.25) can be expressed in terms of the derivatives of the electric fields on the SRRarray. Then, after some manipulations, the following equations are obtained for theelectric fields of the reflected and transmitted waves at z ¼ 0:

Ex

Ey

� �r

¼ A� Yxx B� YxyB� Yyx C � Yyy

� ��1

� Yxx YxyYyx Yyy

� �� Ex

Ey

� �i

(5:28)

Ex

Ey

� �t

¼ A� Yxx B� YxyB� Yyx C � Yyy

� ��1

� A BB C

� �� Ex

Ey

� �i

, (5:29)

where A, B, and C are given by

A ¼ � 2vm0

k2x þ k2zkz

, (5:30)

B ¼ � 2vm0

kxkyk0

; k0 ¼ vffiffiffiffiffiffiffiffiffiffi10m0

p, (5:31)

FIGURE 5.3 Measured transmission coefficient through a CSRR (solid line) and an SRR(dashed line) square planar array. The CSRRs and SRRs are etched on a commerciallow-loss microwave circuit board with permittivity 1 ¼ 2:4310, thickness h ¼ 0.49 mm, andmetallizations of copper with a thickness t ¼ 35mm. The CSRR and SRR parameters arerext ¼ 3.5 mm, c ¼ 0.4 mm, and d ¼ 0.4 mm. The CSRRs and SRRs are arranged in a cubiclattice with periodicity a ¼ 8 mm. The characteristics of the incident waves are described inthe text. (Source: Reprinted with permission from [3]; copyright 2004, the AmericanPhysical Society.)


and

C ¼ � 2vm0

k2y þ k2zkz

, (5:32)

with

kx ¼ kix ¼ krx ¼ ktx (5:33)

ky ¼ kiy ¼ kry ¼ kty (5:34)

and

kz ¼ kiz ¼ �krz ¼ ktz: (5:35)

From equations (5.28) and (5.29), the transmission and reflection coefficients for theco- and cross-polarization components can be obtained for any polarization and angle

of incidence, ui ¼ tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk2x þ k2y Þ=kz

qn o, of the incoming wave.

The main effects that appear for oblique incidence are analyzed in [4], wherean experimental confirmation of these effects is also provided. From equations(5.28)–(5.35), it follows that strong reflections peaks will appear in the SRR arrays(for the appropriate polarization of the incident waves) near the frequency ofresonance of the SRRs. This independence of the surface resonances of the angleof incidence is a remarkable property of the analyzed SRR/CSRR arrays.Regarding cross-polarization effects, from equations (5.28)–(5.35) it follows thatcross-polarization effects are present in general. However, for the two orthogonalmain planes of incidence (i.e., the x2 z and the y2 z planes), from equations(5.28)–(5.35) it follows that cross-polarization only occurs if Yyx or Yxy arenonzero. That is, cross-polarization effects do not appear for incidence in the x 2 zplane. The experimental results for transmittance through the SRR array already con-sidered in Figure 5.3, corresponding to a TE wave with this kind of incidence (or TEy

wave3), are shown for several angles of incidence in Figure 5.4a (the transmittance fora TMy wave is, as expected, near unity for all angles of incidence). The experimentalresults confirm the absence of cross-polarization effects, as well as the presence of ahigh reflectivity around the frequency of resonance of the SRRs. From the Babinetprinciple, a complementary behavior is expected for TMy waves incident upon aCSRR array. This behavior is shown in Figure 5.4b.

For incidence not in the x2z plane, cross-polarization effects are unavoidable.This effect can be seen in Figure 5.5a, where the measured transmittances forplane TMx waves incident upon the same SRR array are shown. These experimentalresults confirms the presence of strong cross-polarization effects near the SRR reson-ance. The same effect can be observed in Figure 5.5b, where the transmittance for the

3In our notation, the subindex, x or y, indicates the polarization of the main field component (electric for TEwaves and magnetic for TM waves), which should also be perpendicular to the plane of incidence.


complementary TEx waves incident upon the CSRR array are shown. From thegeneral form of the cross-polarization aem

yz , it follows that for loss-less arrays Yxy isimaginary, and Yxx, Yyy are real quantities. Therefore, from equations (5.28)–(5.35), it follows that the co- and cross-polarized transmitted (or reflected) wavesmust be in quadrature.4 That is, when cross-polarization is present, the transmittedand reflected waves are elliptically polarized. When both components are of equalmagnitude, the transmitted or reflected wave is circularly polarized. This possibilityis confirmed by the experimental results shown in Figure 5.5 for both arrays.

FIGURE 5.5 Measured transmittance through the SRR (a) and the CSRR (b) arrays ofFigure 5.3 for incidence in the y�z plane at several angles. Incident waves are TMx for theSRR array and TEx for the CSRR array. (Source: Reproduced with permission from [4]; copy-right 2006, Taylor & Francis Group, LLC.)

FIGURE 5.4 Measured transmittance through the SRR (a) and the CSRR (b) arrays ofFigure 5.3 for incidence in the x�z plane at several angles. Incident waves are TEy for theSRR array and TMy for the CSRR array. (Source: Reproduced with permission from [4]; copy-right 2006, Taylor & Francis Group, LLC.)

4Note that for this incidence B ¼ 0.


SRR and CSRR arrays belong to a family of planar devices presenting enhancedtransmission and/or reflection in some frequency band. As has already been men-tioned, the transmission and reflection properties of SRR/CSRRarrays do not cruciallydepend on periodicity. Therefore their behavior is closer to a frequency-selectivesurface than to a diffraction screen. In addition, their transmission and reflectionproperties—which, in some cases, may include polarization conversion—stronglydepend on the polarization of the incident wave. Such a high variety of effectspresent in SRR/CSRR surface arrays may find application in devices such as artificialimpedance and/or frequency-selective surfaces, as well as polarizers and/or polari-zation converters. As it will be shown in Section 5.4, they may also find applicationin the design of microwave lenses with enhanced resolution.

5.3 MAGNETO- AND ELECTRO-INDUCTIVE WAVES

Although some historical precedents can be found (see [7] and references therein)magneto-inductive waves (MIWs) were first reported and systematically studied inthe frame of metamaterial science in [8,9], and subsequent works from the samegroup. Magneto-inductive waves were first reported in one-, two-, and three-dimensional arrays of capacitively loaded metallic rings. After such seminal works,MIWs were soon reported in arrays of swiss rolls [11]5 and SRRs [10]. Magneto-inductive waves are quasimagnetostatic waves, where the field behavior is mainlygoverned by Ampere’s and Faraday’s laws. Therefore, the effects of the displacementcurrent—for instance, retardation—are ignored. As usual in quasistatic waves,6 thisapproach is only valid in the short wavelength limit (k � k0 ; v


p).

Therefore, MIWs are slow waves, with phase and group velocity usually muchsmaller than the velocity of light in free space. In bulk (three-dimensional)SRR arrays, MIWs can be considered as a special kind of magnetostatic wave,arising when the internal wavelength is of the same order as the interspacingbetween the SRRs. In fact, if the bianisotropic effects are ignored (or nonbianiso-tropic elements, such as BC-SRRs or NB-SRRs, are used) and the internal wave-length is of the same order as the SRRs’ interspacing, it can be expected thatinteractions between nearest neighbors through Faraday’s law dominate. In suchcase we are in the realm of the magneto-inductive approximation. Thus, we candistinguish between three kinds of waves in three-dimensional SRR arrays:

† Electromagnetic waves, where the internal wavelength is large with regard toSRR interspacing and of the same order as the free-space wavelength;

5Swiss rolls were introduced as a mechanism to obtain magnetic flux channelization in the megahertz range[12] (see also Section 5.4).6Other examples of quasistatic waves are surface plasmons in metallic interfaces [13] or magnetostaticwaves in ferrite samples [14,15].


† Conventional magnetostatic waves (see Problem 2.10), where the internal wave-length is large with regard to SRR interspacing, but short with regard to thefree-space wavelength;7 and

† MIWs, where the internal wavelength is of the same order as SRR interspacing,and short with regard to the free-space wavelength.8

Electro-inductive waves (EIWs) can be considered as the quasielectrostaticcounterpart of magnetoinductive waves. In such a case, the wave equation is domi-nated by the quasielectrostatic interaction (Coulomb’s law) between the elementsof the structure. Electro-inductive waves in chains of metallic resonators havebeen known for a long time [16]. More recently, the interest in this kind of quasielec-trostatic waves has been renewed in connection with the problem of the generationand guidance of electromagnetic slow waves at optical frequencies [17,18]. In thissection, we will restrict our analysis to the electro-inductive waves guided by one-and two-dimensional arrays of CSRRs [19]. Such specific EIWs can be consideredas the dual counterpart of the aforementioned magneto-inductive waves inSRR arrays.

5.3.1 The Magneto-Inductive Wave Equation

Let us consider the one-dimensional chains of conducting NB-SRRs9 shown inFigure 5.6. Considering loss-less rings and neglecting other than nearest-neighborinteractions, the current on the nth NB-SRR, In, is given by

jvLþ 1jvC

� �In ¼ �jvM In�1 þ Inþ1ð Þ, (5:36)

where L and M are the self and mutual inductances of the NB-SRRs, and C is theNB-SRR total capacitance.10 The dispersion equation for MIWs propagating alongthe chain is computed from equation (5.36) assuming a current dependence of thekind In ¼ exp{�jk(m� n)a}Im, where k is the propagation constant and a the

7In fact, in many practical SRR arrays, where the electrical size of the elements is �1/10 of the free-spacewavelength, this region may be very small or irrelevant.8Therefore, the magneto-inductive approach can be very useful for the analysis of waves propagating inbulk SRR-based metamaterials in the short wavelength region (see Section 2.4.3 of Chapter 2).9This particular configuration has been chosen for simplicity. However, a similar analysis will hold forother SRR designs such as BC-SRRs or EC-SRRs (provided cross-polarization effects are not present,or can be neglected), as well as for similar arrays of capacitevely loaded rings, metallic swiss-rolls, andother similar elements.10That is, C ¼ prCpul/2, where Cpul is the per-unit-length capacitance in the slot between the rings, and r isthe average radius of the NB-SRR (see Chapter 2).

5.3 MAGNETO- AND ELECTRO-INDUCTIVE WAVES 279

periodicity of the array. The final result is [9]

v20

v2¼ 1þ 2M

Lcos(ka), (5:37)

where v0 is the frequency of resonance of the isolated NB-SRR

v20 ¼

1LC

¼ 2prCpul

: (5:38)

The parameters L and C appearing in equation (5.37) can be calculated following theprocedure developed in Section 2.3.1. The mutual inductance is more difficult toevaluate in a closed form, because it depends on the specific disposition of theNB-SRRs. Losses in magneto-inductive waveguides are mainly related to ohmiclosses in the resonators. They can be easily taken into account (see Problem 5.4)by including the ring ohmic resistance in the current equation (5.36).

The group velocity associated with the dispersion relation (5.37) is

vg ;@v

@k¼ M

Lv0a sin(ka) 1þ 2

M

Lcos(ka)

� ��3=2

: (5:39)

It can be easily realized by inspection that the mutual inductance,M, is positive in theaxial configuration and negative in the coplanar configuration of Figure 5.6.Therefore, MIW propagation is forward in the axial configuration, and backward inthe co-planar configuration. The bandwidth for the excitation of MIWs can also bededuced from equation (5.37). Assuming, as usual, that jM=Lj � 1, it iseasily deduced from equation (5.37) that MIWs can be excited in the frequency

FIGURE 5.6 One-dimensional chain of NB-SRRs of periodicity a. Axial configuration (a)and coplanar configuration (b).


interval v0 � dv , v , v0 þ dv, with

dv ¼ v0jMjL

: (5:40)

The dispersion equation (5.37) can be easily generalized to two- and three-dimensional arrays of SRRs and similar elements. For a three-dimensional cubicarray of NB-SRRs, similar to that shown in Figure 2.12, considering only interactionsbetween nearest neighbors, the dispersion relation is

v20

v2¼ 1þ 2Mc

Lcos(kxa)þ cos(kya)� �

þ 2Ma

Lcos(kza), (5:41)

whereMc andMa are the mutual inductances in the coplanar and axial configurations,respectively. The magneto-inductive wave equation can also be generalized to thecase when the nearest-neighbor approximation is not enough to characterize thesystem. If the interactions between all the elements are taken into account, themagneto-inductive wave equation (5.41) becomes

v20

v2¼ 1þ

Xl

Xn

Xm

2Mlnm

Lcos a(lkx þ nky þ mkz)

� , (5:42)

where the notation is rather obvious. Regarding the usefulness of the nearest-neighborapproximation, it becomes less and less accurate when the dimensionality of thesystem increases. In fact, for rings located beyond some critical distance, thedipolar approximation must be valid. For such rings the mutual inductances decayas 1/d3, where d is the distance to these rings. However, the number of ringslocated at such a distance varies as 1, d, and d2 for one-, two-, and three-dimensionalconfigurations, respectively. Therefore, the influence of the rings located far awayfrom a given ring substantially increases with the dimensionality of the configuration.The usefulness of the nearest-neighbor approximation also depends on the specificcharacteristics of the elements.11

The coupling between magneto-inductive and electromagnetic waves in a compo-site including SRRs has also been investigated [20]. In this analysis such coupling ismodeled as the coupling between a chain of SRRs and a transmission line. The equiv-alent circuit proposed in [20] is similar to the circuit model reported in Figure 3.23,but also including the magnetic coupling between nearest SRRs through a mutualinductance. As this circuit model implies a nearest-neighbor approximation, the accu-racy of the reported model is higher for shorter internal wavelengths. Thus, for left-handed composites, the model is expected to provide better accuracy at the lowerend of the left-handed pass band. For the same reason, better accuracy is expectedfor low-dimensional structures (one- and two-dimensional structures).

11For instance, it is not appropriate for one-dimensional chains of swiss rolls [11].


5.3.2 Magneto-Inductive Surfaces

Two-dimensional square planar arrays of NB-SRRs and similar configurationssupport magneto-inductive waves, which, in the nearest-neighbor approximation,satisfy the equation

v20

v2¼ 1þ 2Mc

Lcos(kxa)þ cos(kya)� �

: (5:43)

When the incidence of plane waves on the array is analyzed, solutions to equation(5.43) must appear as poles of the transmission and reflection coefficients for imaginaryvalues of the transverse wavevector kz (see Chapter 1). This effect cannot be taken intoaccount by the simple homogenization procedure used in Section 5.2.2. Throughoutthis section, we will develop a modification of such homogenization, which takesinto account the excitation of MIWs in the surface array. This modified formalism isbased on the analysis developed in [22] for plane arrays of coupled resonators.

For simplicity, a plane array of NB-SRRs will be considered. Therefore, cross-polarization effects are not present, a fact that substantially simplifies the analysis.The size of the NB-SRRs, as well as the periodicity of the array will be assumedelectrically small. Therefore, the admittance approximation (5.10) holds. However,because the NB-SRRs do not exhibit cross-polarization effects, the array is isotropicin the x–y plane. Therefore, equation (5.10) is substituted by the simpler equation

Js ¼1Za

Ek, (5:44)

where Za is the array impedance and Ek is the component of the average electric fieldparallel to the array. A cell impedance, Zc, is now defined. This impedance relates thesurface current on the array with the field of the incident wave

Js ¼1Zc

Eik, (5:45)

where Eik is the component of the incident field parallel to the array. The average field

Ek is related to the incident field by

Ek ¼ Eik �

12Z0Js, (5:46)

where the second term accounts for the electric field of the TE wave created by thesurface currents on the array. In this equation Z0 is simply the wave impedance forTE waves,

Z0 ¼vm0

kz; kz ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv210m0 � k2x � k2y

q(5:47)


with Im(kz) , 0. In the quasistatic approximation, this expression reduces to

Z0 ¼jvm0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y

q : (5:48)

Equations (5.46)–(5.48) are the basis for the new homogenization procedure. Thecell impedance Zc can be calculated from the NB-SRR polarizabilities as will beshown later, and the array impedance is obtained from equations (5.44)–(5.46):

Za ¼ Zc �12Z0: (5:49)

The average surface current on the array can be obtained from equation (5.11) withPs ¼ 0 (cross-polarization effects are not present) andMs¼ mza�2 z, where mz is themagnetic moment induced in the NB-SRRs. This magnetic moment can be obtainedfrom the external magnetic field, Bext ¼ jr� Eext=v, seen by each NB-SRR through(see Chapter 2)

mz ¼ aBextz ; a ¼ p2r4

L

v20

v2� 1

� ��1

, (5:50)

where r is the mean radius of the NB-SRR. In the nearest-neighbor approximation,this external magnetic field can be expressed as

Bextz ¼ Bi

z þ2Mp2r4

cos(kxa)þ cos(kya)� �

mz: (5:51)

In this equation, Bz is the magnetic field of the incident wave, and the last termaccounts for the magnetic flux through the NB-SRR due to its four nearest neighbors.Putting all this together, the cell impedance (5.45) is found as

Zc ¼jvm0a

2

k2x þ k2y

1m0a

� b

� �, (5:52)

where

b ¼ M

2m0p2r4

cos(kxa)þ cos(kya)� �

: (5:53)

It can be realized from this expression that the MIWs supported by the array (5.43)correspond to the zeroes of the cell impedance Zc. That is, they correspond to a situ-ation on which, according to equation (5.45), there is a nonvanishing surface current


on the array in the absence of any incident wave. Finally, from equations (5.49)and (5.52), it is concluded that the surface impedance of the considered array ofNB-SRRs is

Za ¼jvm0a

2

k2x þ k2y

1m0a

� b

� �� 12Z0: (5:54)

The transverse transmission matrix of the NB-SRR array can be defined as inChapter 1,

Eþ1

E�1

� �; T

¼� Eþ

2E�2

� �¼ T11 T12

T21 T22

� �� Eþ

2E�2

� �, (5:55)

where E+1 (E+

2 ) is the electric field for the positive/negative waves at the left-(right-)hand side of the array (see Chapter 1 for more details on the notation). This matrix canbe obtained from the boundary conditions at both sides of the array (continuity of thetangential electric field and Ampere’s law for the magnetic field), and from equation(5.44). The final result is12

T¼¼ 1

22þ Z0=Za Z0=Za�Z0=Za 2� Z0=Za

� �: (5:56)

Magneto-inductive waves correspond to solutions of equation (5.55) withEþ1 ¼ E�

2 ¼ 0. That is, T11 ¼ 0 or Za ¼ �Z0=2. From equation (5.54) it can beeasily realized that this condition is actually fulfilled by MIWs with the dispersionrelation (5.43).

Although the above analysis has been carried out for a particular kind of element(the NB-SRRs), it is also valid for any array of magneto-inductive rings, providedcross-polarization effects can be neglected and the nearest-neighbor approximationcan be considered valid. In fact, the extension of this analysis to situations wherethe aforementioned conditions are not fulfilled is cumbersome but straightforward.

5.3.3 Electro-Inductive Waves in CSRR Arrays

As has already been mentioned, propagation of electromagnetic waves through chainsof electrically coupled resonators has long been known at microwave frequencies[16], and more recently at optical frequencies [17,18]. In this section we willaddress the propagation of such types of waves in chains of CSRRs. For simplicity,it will be assumed that cross-polarization effects are not present or can be neglected.Figure 5.7a shows an example of such a kind of chain, whose unit cell is acomplementary NB-SRR (C-NB-SRR) etched on a metallic plate. As was shownin Chapter 3, these resonators behave as an LC circuit, presenting a strong electricpolarizability at resonance. Therefore, the coupling between the elements of the

12There is a small difference regarding the expression reported in [22] due to the different definitions for thetransmission matrix.


chain is capacitive. The equivalent circuit for the unit cell is shown in Figure 5.7b,where CG is the capacitance to ground of the internal disk of the C-NB-SRR, andLC the self-inductance of the resonator.13 The mutual capacitance CM is the capaci-tance between the inner disks of two consecutive C-NB-SRRs. Considering aninfinite chain, the dispersion equation arising from this equivalent circuit is [19]

v2

v20

¼ 1þ 2v2CMLC cos(ka), (5:57)

where

v20 ¼

1(CG þ 2CM)LC

: (5:58)

As CG þ 2CM ¼ CC is approximately the total capacitance between the inner diskof an isolated C-NB-SRR and the ground, this v0 is approximately the frequencyof resonance of an isolated C-NB-SRR. The C-NB-SRR chain is, in fact, the dual ofthe coplanar NB-SRR chain shown in Figure 5.6b. Therefore, the dispersion relationmust be the same for both configurations (see Problem 5.6).

The same considerations made in Section 5.3.1 for two-dimensional arrays of NB-SRRs (it is apparent that three-dimensional arrays of C-NB-SRRs cannot be made)can be extended to two-dimensional arrays of C-NB-SRRs (see Problem 5.7). Thecoupling between CSRRs and transmission lines, taking into account the mutualcapacitive couplings between the nearest CSRRs, has been studied in [23]. Thisanalysis can be considered as the dual counterpart of the analysis developed in [20].

5.3.4 Applications of Magneto- and Electro-Inductive Waves

Magneto-inductive waves have potential applications in radio-frequency (RF), micro-wave, and millimeter-wave applications, that is, from the megahertz to the terahertzrange. Such applications may take advantage of the versatility and simplicity of MIWdevices, as well as from its relatively easy mathematical analysis. Magneto-inductivewaveguides and devices, including bends, power dividers, couplers, and phase shifters,

FIGURE 5.7 Chain of complementary NB-SRRs (C-NB-SRRs) etched on a metallic plate(left) and equivalent circuit for the unit cell (right).

13As was shown in Chapter 3, this self-inductance comes from the connection in parallel of the inductancesof the co-planar waveguide formed between the inner disk of the C-NB-SRR and the ground.


have been demonstrated and described in [24] and [25] among others (see also [7] andreferences therein). As an example, Figure 5.8 reproduces the design of a powerdivider and a backward-wave coupler in MIW technology.

Applications of MIWs in conventional microstrip microwave technology have alsobeen proposed. Transduction from conventional microstrip to planar magneto-inductive waveguides is possible by simply etching the magneto-inductive waveguideon the microstrip dielectric substrate [10].14 A five-step microstrip MIW transducer isshown in Figure 5.9. This device provides a transmission coefficient of 25 dBbetween the input and output microstrips, with a bandwidth of 0.5 GHz centered at4.5 GHz. As has already been mentioned, MIWs are slow waves, which can beexcited in a narrow frequency band. Such properties strongly suggest the usefulnessof the reported MIW transducers as microwave filters and/or delay lines. Specifically,they have potential interest as an alternative to surface acoustic wave and/or ferritedelay lines. The time delay provided by the microstrip MIW transducer ofFigure 5.9 is also shown in the figure. It corresponds to a slow-wave factorc=vg � 100. This value, although high, is still smaller than those provided by conven-tional surface acoustic or ferrite delay lines. However, MIW transducers have a muchsimpler design and, due to the scalability of the SRRs, they can be designed over amuch wider bandwidth (from RF to terahertz). Electro-inductive wave transducersin microstrip technology, with performances quite similar to the aforementionedmicrostrip MIW transducers, have also been reported [19].

FIGURE 5.8 Schematic of a power divider (a) and a coupler (b) in planar magneto-inductivewaveguide technology. (Source: Reprinted with permission from [7], copyright 2006, Elsevier.)

14Care must be taken, however, in placing the transition from the microstrip to the magneto-inductive wave-guide near the region where current in the microstrip has a maximum, and in such a way that the magneticflux lines are properly coupled to the magneto-inductive elements. This configuration can be seen inFigure 5.9, where the first SRR is placed at a quarter-wavelength from the microstrip open circuit.


Finally, magneto-inductive devices may also find interesting applications in RF andmicrowave imaging devices [12,26,27]. Wewill return to this topic in the next section.

5.4 SUBDIFFRACTION IMAGING DEVICES

After the seminal paper of Pendry [28] on the ability of left-handed slabs to overcomethe diffraction limit for lens resolution (see Chapter 1), this issue became one of the

FIGURE 5.9 Photograph (a), sketch (b), and delay time (c) provided by the planar MIWtransducer proposed in [10]. The input and output ports are conventional microstrip lines,and MIWs are guided by a chain of modified NB-SRRs of rectangular shape (in order toenhance mutual inductance) etched on the dielectric substrate. (Source: Reprinted with per-mission from [10], copyright 2004, American Institute of Physics.)

5.4 SUBDIFFRACTION IMAGING DEVICES 287

most active research topics in metamaterial science. It was already noted by Pendry inhis seminal paper that, for near-field subdiffraction imaging, it is not necessary tohave a left-handed slab, but simply a slab with only negative permittivity (for TMpolarization). This was the first proposal of a device different from the left-handedperfect lens (see Chapter 1) able to produce subdiffraction imaging. Other proposalscame soon after, such as imaging by planar circuit arrays [29], imaging by anisotropicmedia with negative parameters or indefinite media [30], ferrite slabs [31], coupledarrays of weakly coupled electric resonators [22], coupled magneto-inductive sur-faces [27], and plane slabs of two- and three-dimensional photonic crystals[32,33]. Although this list is probably incomplete, it gives an idea of the intenseresearch activity on this topic. Among the motivations of such proposals was thedesire to find more practical devices, easier to manufacture than a left-handed slab.In this section, we will examine some such proposals, with emphasis on potentialpractical advantages. However, before developing this analysis, some general charac-teristics of any subdiffraction imaging device will be analyzed.

In addition to the above devices, which can create subdiffraction images of sourceslocated at some distance from the lens, some metamaterial structures can producepixel-to-pixel image translation between front- and back-side interfaces. Someexamples of such devices are the hexagonal swiss-roll structure proposed in [12],the wire media proposed in [34], and the photonic band gap collimation effect ana-lyzed in [35]. This image translation is based on mechanisms quite different fromthat described in Chapter 1 for Pendry’s super-lens. Such devices, which can bemore properly termed as canalization devices, will be briefly analyzed at the endof this section.

5.4.1 Some Universal Features of Subdiffraction Imaging Devices

The main characteristics of subdiffraction imaging by left-handed slabs have alreadybeen reported in Chapter 1, and include

1. Field decay at the backward side of the lens, which implies that there is nofocusing of energy in spots of subdiffraction size;

2. Matching capabilities, which imply that some apparent three-dimensionalfocuses may appear in the image detection process, as a consequence of thetunneling of power between the source and the detector;

3. High sensitivity to material losses, which restricts the usefulness of suchdevices to near-field imaging.

Although the analysis in Chapter 1 was mainly focused on left-handed slabs, it can beshown that the above properties are actually independent of the specific nature of thelens, thus being general characteristics of any subdiffraction imaging device [36]. Infact, regarding the first property, this comes directly from the fact that the transferfunction in free space for any evanescent spatial Fourier harmonic must be an expo-nential decaying function, regardless of the specific nature of the lens. Therefore,the restoration of evanescent spatial Fourier harmonics at some distance from the


back-side interface of the lens, implies necessarily the presence of fields decayingfrom this interface towards the image.

In order to illustrate such an effect, let us consider a hypothetical optical system(see Fig. 5.10) able to reproduce some time–harmonic excitation field imposed atsome plane (the source plane at x ¼ 0 in Fig. 5.10) at some other plane (the imageplane, at x ¼ xi in Fig. 5.10) with a resolution D. Such an optical system is locatedbetween x1 and x2 (0 , x1 , x2 , xi). For simplicity, the excitation field will beassumed to be a delta function in the x ¼ 0 plane

ce(0, y, z) ¼ (2p)2d( y, z), (5:59)

where ce is some component of the electric/magnetic field tangential to the plane x ¼0 (for simplicity, the remaining tangential component of the excitation field is sup-posed to vanish at x ¼ 0). The total field for x. 0 will be the superposition of theexcitation field and the scattered field created by the optical system. For x. 0 theexcitation field can be expanded as a Fourier integral of plane waves:

ce(x, y, z) ¼ð1�1

dky

ð1�1

dkz e�jkxx�jkyy�jkzzþjvt, (5:60)

where kx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv210m0 � k2y � k2z

q. In order to have physical fields, the sign of the

square root has to be chosen with Re(kx) . 0, Im(kx) ¼ 0 for propagative waves,

FIGURE 5.10 Illustration of the image formation by an optical device. A point excitation islocated at z ¼ 0, creating the excitation field ce(0, y, z) ¼ (2p)2d( y, z) at such a plane. Theimaging device, located between x ¼ x1 and x ¼ x2, restores the excitation field at the imageplane with a resolution D. As a result, a two-dimensional spot of size �D is created at theimage plane. Outside this spot, the field rapidly vanishes in the x ¼ xi plane.


and with Re(kx) ¼ 0, Im(kx) , 0 for evanescent waves. Because the optical system issupposed to reproduce the excitation field at x ¼ xi with a resolution D, the total fieldat x ¼ xi must be

c(xi, y, z) �ðkmax

�kmax

dky

ðkmax

�kmax

dkz e�jkyy�jkzzþjvt , (5:61)

where the resolution D is given by

kmax ¼ 2p=D: (5:62)

As there are no sources for x . x2, the total field for x2 , x , 1 is the analyticalcontinuation of equation (5.61); that is

c(x, y, z) �ðkmax

�kmax

dky

ðkmax

�kmax

dkz e�jkyy�jkzz�jkx(x�xi)þjvt: (5:63)

If D , l ¼ 2p=k0, some of the plane waves included in equation (5.63) are evanes-cent, decaying from x ¼ x2 towards x!1. Figure 5.11 shows several field distri-butions around x ¼ xi, corresponding to some values for the resolution D. The fielddistribution is calculated from equation (5.63) with kmax given by equation (5.62).The focusing of energy in a three-dimensional spot at the image can be clearlyseen when D . l. However, a field decay from the lens towards the image can beclearly observed when D, l. Such field decay is compatible with image formationat the image plane (x ¼ xi) with a resolution D, according to equation (5.61).

Regarding the second characteristic of the above list, this comes directly from thefact that evanescent electromagnetic waves do not carry power. Therefore, any detec-tion process, which actually implies some amount of power transmitted between thesource and the detector, must imply a substantial perturbation of the field created bythe source in the presence of the lens alone. This detection process was actually ana-lyzed in Sections 1.9.2 and 1.9.4, and the main conclusions of such an analysis can beextended without modification to other subdiffraction imaging devices. Therefore,care must be taken in any image measurement in order to avoid artifacts arisingfrom the matching capabilities of such devices [37,38]. The presence of apparentthree-dimensional focusing in imaging measurements can be one such artifact [39].To avoid them, probes that do not substantially affect the field at the back side ofthe lens must be used. Small electric dipoles and/or magnetic loops, as well as resis-tive loads in the receiving antenna can be used for this purpose [36,39]. On the otherhand, future practical applications based on such matching capabilities of subdiffrac-tion imaging devices can also be envisaged.

The high sensitivity of subdiffraction imaging devices to material losses alsocomes from the exponential decay of the field at the back side of the lens. This


field decay implies that, in order to have a good image at some distance from the backinterface of the lens, very high densities of energy must be present just at this inter-face. As material losses are proportional to the field intensity, this behavior impliesvery high losses inside the lens (see also Section 1.10). In order to avoid such

FIGURE 5.11 The total field intensity jc(x, y, z)j obtained from equation (5.63) at both sidesof the image plane x ¼ xi, for several values of the resolution D: (a) D ¼ 3l; (b) D ¼ l; (c)D ¼ l=3. (Source: Reprinted with permission from [36], copyright 2006, American Instituteof Physics.)


losses, it is better to place the lens in such a way that the image is formed near its backinterface.15 Due to this high sensitivity to losses, subdiffraction imaging devices are,in practice, near-field devices. To overcome losses in such devices, multilayer con-figurations have been proposed [41,42]. However, it must be clarified that suchproposals, although enlarging the distance between the source and the image, donot enlarge the available distances between the image (or the source) and the lens.

5.4.2 Imaging in the Quasielectrostatic Limit: Role ofSurface Plasmons

Because metamaterial super-lenses are near-field devices, it may be of interest that theanalysis of quasistatic devices is able to produce similar effects. In fact, quasistaticapproaches are approximately valid when the scale of the spatial variations of thefield is small with regard to the free-space wavelength, which is, indeed, the maincharacteristic of near-field devices. In the quasielectrostatic limit E ¼ �rf, wheref is the quasielectrostatic potential. Quasielectrostatic plane waves are TM waves,which, in a homogeneous and isotropic medium, satisfy Laplace’s equationr2f ¼ 0. Surface plasmons (see Chapter 1, Problems 1.2 and 1.3) are examplesof such waves. In order to develop our analysis, it will be of interest to determinethe quasielectrostatic transverse transmission matrix (see Sections 1.7 and 1.8) foran air–dielectric interface, and for a dielectric slab in air. According to the definitionsof Chapter 1, TM quasielectrostatic positive/negative waves of the formf+ ¼ f0 exp{�jkz+ ax} will be considered, where, from Laplace’s equation,

a ¼ jkj: (5:64)

The transverse transmission matrix for a dielectric interface perpendicular to thex-axis is defined as

fþ1

f�1

� �; T

¼� fþ

2f�2

� �¼ T11 T12

T21 T22

� �� fþ

2f�2

� �, (5:65)

where f+1 (f+

2 ) stand for the potentials at the left-(right-)hand side of the interface.This matrix is readily calculated from the boundary conditions at the dielectric inter-face: continuity of the quasistatic potential f ¼ fþ þ f� and continuity of thenormal component of the field displacement vector Dx ¼ 1(afþ � af�). Theresult is

T¼d ¼

1211

11 þ 12 11 � 1211 � 12 11 þ 12

� �, (5:66)

15In fact, it has been reported that the best resolution is obtained when the image is formed just on thisinterface [40].


where 11(12) is the dielectric constant at the left-(right-)hand side of the interface.From this result, the transverse transmission matrix for the dielectric slab ofFigure 5.12 (from x1 to x2) can be easily found:

T¼s ¼

14101

10 þ 1 10 � 1

10 � 1 10 þ 1

� �� ejkjd 0

0 e�jkjd

� �� 1þ 10 1� 10

1� 10 1þ 10

� �: (5:67)

For the specific case when 1 ¼ �10, which corresponds to the condition for exci-tation of surface plasmons at an air–slab interface (see Chapter 1, Problem 1.2),this matrix becomes

T¼s ¼ e�jkjd 0

0 ejkjd

� �: (5:68)

Thus, the quasistatic transverse transmission matrix between x ¼ 0 and x ¼ 2d inFigure 5.12 is the identity

T¼

0!2d ¼1 00 1

� �: (5:69)

That is, there is perfect matching, as well as amplitude and phase compensation for allthe TM quasielectrostatic waves between x ¼ 0 and x ¼ 2d. Therefore, for 1 ¼ 210the quasielectrostatic field at x ¼ 0 is reproduced at x ¼ 2d, which is the condition forperfect lens behavior. Moreover, from equations (5.67)–(5.69), it follows that theamplitude of the quasielectrostatic potential has the decaying-growing-decayingbehavior shown in Figure 5.13, just as in the left-handed perfect lens analyzed inChapter 1. However, it must be taken into account that the quasielectrostatic analysisdeveloped in this section is approximate, and valid only for jkj � k0 ; v


p.

FIGURE 5.12 A slab of nonmagnetic medium of width d.


As has already been mentioned, quasistatic subdiffraction imaging by negativepermittivity media was first proposed in [28] as a useful alternative to left-handedperfect lenses at infrared and optical frequencies. In fact, the complex permittivityof metals is mostly real and negative at such frequencies [2]. Experimentalconfirmation of such an effect in thin silver slabs has been provided in [43] andin [44].

As the above analysis find its main application in imaging by thin metallic slabs, adeeper analysis of the transfer function when the slab is made of a medium thatsatisfies a Drude model for its permittivity is of interest. The lens transfer functionis defined as the transmission coefficient between x ¼ 0 and x ¼ 2d of Figure 5.12,which can be deduced from equation (5.67) as

T(jkj,v) ¼ e�ad

[Ts]11¼ 4101e�jkjd

(1þ 10)2ejkjd � (1� 10)2e�jkjd , (5:70)

which takes the desired value of T ¼ 1 when 1 ¼ 210. For our purposes it is con-venient to rewrite this transfer function, as well as the complex permittivity of theslab (Eq. 2.78 of Chapter 2), in terms of the dimensionless variables k0 ¼ jkjd andv0 ¼ v=vp, and the dimensionless parameter f 0 ¼ fc=vp:

T(k0,v0) ¼ 4101e�k0

(1þ 10)2ek0 � (1� 10)2e�k0

, (5:71)

where

1 ¼ 10 1� 1v0(v0 � jf 0)

� �: (5:72)

FIGURE 5.13 Variation along the lens of the amplitude of the quasielectrostatic potential ina quasielectrostatic quasiperfect lens.


A plot of T(k0,v0) for f 0 ¼ 10�2 (which can be considered a typical value for goodconductors) is shown in Figure 5.14. The two crests that appear in this plot corre-spond to the surface plasmons supported by the slab (see Chapter 1). Betweenthese crests there is a region where the transfer function is flat and approximatelyequal to unity. This is the region of the (v0, k0) plane for which the restoration ofspatial Fourier harmonics takes place. Maximum values of k0 inside this region(i.e., better resolution) are obtained for v0 ¼ 1=

ffiffiffi2

p, which corresponds to the fre-

quency of excitation for the quasistatic surface plasmons at the interface of a semi-infinite slab. As resolution is limited by the maximum value of k for which T � 1,it is clear from Figure 5.14 that resolution is better for thinner slabs (k ¼ k0=dreaches higher values for thinner slabs). However, it is also clear from the figurethat, for very high values of k0 ¼ kd, the bandwidth of both surface plasmons col-lapses to a single band, and the transfer function cannot reach values near unity.Thus, it can be concluded that super-resolution in plasmonic super-lenses islimited to that region of the (v, k) plane where two different surface plasmons canbe clearly identified in the slab, and disappears when the bandwidth of bothsurface plasmons collapse to a single frequency band.

5.4.3 Imaging in the Quasimagnetostatic Limit: Role of MagnetostaticSurface Waves

Quasimagnetostatics, like quasielectrostatics, can also be applied to the analysisand design of near-field devices. It is apparent, from duality, that a slab with1 ¼ 10 and m ¼ �m0, (which could be made from an appropriate array of smallsize SRRs) can also produce subdiffraction imaging. However, this possibility doesnot have too much practical interest because—unlike negative-1 media—such

FIGURE 5.14 Plot of the transfer function (5.71) as a function of the dimensionless vari-ables k0 and v0, for f 0 ¼ 10�2. The values for v0 are centered around v0 ¼ 1/

p2 which corre-

spond to Re(1) ’ �1.


kinds of materials are not directly available in nature. Of much more interest is findingand analyzing quasimagnetostatic subdiffraction imaging devices that can be madefrom available materials. Because ferrites exhibit regions of negative permeabilityat microwave frequencies (see Chapter 2), they are good candidates for such apurpose. Moreover, from the role played by surface plasmons in quasistaticimaging devices, and the similarities between surface plasmons and magnetostaticsurface waves (MSSWs), it can be guessed that such waves can play a similar rolein quasimagnetostatic subdiffraction imaging. Therefore, we will analyze the trans-mission of quasimagnetostatic waves through the ferrite slab of Figure 5.15, whichhas the proper magnetization for the excitation of MSSWs propagating alongthe y-axis at each interface (see Problems 2.11–2.13 of Chapter 2). In thequasimagnetostatic approximation H ¼ �rc, where c is the quasimagnetostaticpotential, which satisfies the magnetostatic wave equation (see Chapter 2)

r � ��m � rc ¼ 0, (5:73)

where ��m is the Polder tensor,

��m ¼ m0

��mt 00 mz

� �¼ m0

m jk 0�jk m 00 0 1

0@

1A, (5:74)

where the external magnetic field is assumed along the z-axis, H0 ¼ H0 z, and

m ¼ 1þ vMvH

v2H � v2

(5:75)

k ¼ vvM

v2H � v2

(5:76)

(see Chapter 2 for more details on these definitions).

FIGURE 5.15 Near-field quasimagnetostatic ferrite lens made of a ferrite slab magnetizedalong the direction parallel to the slab.


As in Chapter 1, we will classify the magnetostatic waves into positive and nega-tive plane waves, with the field dependence c+ ¼ c0 exp{�jky+ ax}, where, fromequation (5.73), a and k are related through

a ¼ jkj: (5:77)

The transverse transmission matrix for such waves can be obtained from the boundaryconditions at the air–ferrite interfaces of Figure 5.15. Such boundary conditions areas follows: continuity of c ¼ cþ þ c� and continuity of Bx ¼ mHx þ jkHy. Whenthis last condition is put in terms of the quasimagnetostatic potential, it takes the form

cþ1 � c�

1 ¼ (mþ jk)cþ2 � (m� jk)c�

2 , (5:78)

where j ¼ +1 for k_ 0. This equation is valid for the air ! ferrite interface ofFigure 5.15 (at x ¼ x1) and, as in the previous section, c+

1 =c+2 stands for potentials

at the left-/right-hand side of the interface. From such boundary conditions, thequasimagnetostatic transverse transmission matrix for an air ! ferrite interface isobtained as

T¼a!f ¼

12

1þ mþ jk 1� mþ jk

1� m� jk 1þ m� jk

� �: (5:79)

The dependence on the sign of k of this matrix is closely related to the nonreciprocityof ferrites [15], and with the fact that MSSWs—which correspond to the zeroes of[Ta!f ]11—can only be excited at an air–ferrite interface for a specific sign of k(see Problem 2.13 of Chapter 2). The quasimagnetostatic transverse transmissionmatrix for the ferrite ! air interface of Figure 5.15 (at x ¼ x2) is

T¼

f!a ¼ T¼a!f

n o�1¼ 1

2m1þ m� jk �1þ m� jk

�1þ mþ jk 1þ mþ jk

� �: (5:80)

Finally, from equations (5.79) and (5.80), the transverse transmission matrix for theferrite slab of Figure 5.15 (between x ¼ x1 and x ¼ x2) can be found from

T¼

¼ T¼a!f � ejkjd 0

0 e�jkjd

� �� T¼

f!a: (5:81)

The frequency of excitation of MSSWs at an isolated ferrite ! air interface mag-netized as the right-hand side interface of the ferrite slab (i.e., such as the interfacelocated at x ¼ x2 in Fig. 5.15) can be obtained from [T f!a]11 ¼ 0. This equationhas two positive frequency solutions. For k, 0 (i.e., for j ¼ 1), the solution isv ¼ vH, which is not a practical solution because it corresponds to the ferrimagnetic


resonance. However, for k . 0 (i.e., for j ¼ �1), equation [T f!a]11 ¼ 0 becomes

1þ mþ k ¼ 0, (5:82)

which corresponds to the frequency

v ¼ vH þ vM=2: (5:83)

At this frequency, the ferrite slab transverse transmission matrix (5.81) becomes

T¼þ ¼

e�jkjd �21þ m

msinh(jkjd)

0 ejkjd

0@

1A, (5:84)

for k. 0 (or j ¼ 21). For k, 0 (or j ¼ 1) it is

T¼� ¼

e�jkjd 0

2 1þ mm sinh(jkjd) ejkjd

0@

1A: (5:85)

In both cases the transmission coefficient is

t ¼ 1=[T+]11 ¼ ejkjd, (5:86)

which corresponds to the condition for the restoration at the image plane (x ¼ 2d) ofthe amplitude of the incident quasimagnetostatic wave at x ¼ 0. This result, which isvalid for any value of k, shows that a properly magnetized ferrite slab can reproducethe incident quasimagnetostatic field at the source plane (x ¼ 0 in Fig. 5.15) at theimage plane (x ¼ 2d in Fig. 5.15).

The main difference between quasimagnetostatic subdiffraction imaging by mag-netized ferrite slabs, and the quasielectrostatic imaging analyzed in the previoussection is the nondiagonal form of the transverse transmission matrices (5.84) and(5.85) for ferrites. This nondiagonal form introduces some key differences. Infact, for k. 0 the reflection coefficient r ¼ [Tþ]12=[Tþ]11 vanishes.16 However, ifk, 0, the reflection coefficient r ¼ [T�]12=[T�]11 is not zero. Because a generalfield distribution at x ¼ 0 must incorporate Fourier harmonics with positive and nega-tive values of k, it is clear that, in general, the ferrite slab will not reproduce at theimage the whole field at the source plane (which is the sum of the incident fieldand the reflected field), in spite of the fact that it reproduces the incident field (i.e.,the field created by the source alone). This behavior makes a substantial differencewith previously reported subdiffraction imaging devices, which are matched to free

16That is, in general, for k/ H0 � n, where n is the outward normal of the back-side lens interface.


space (i.e., do not produce reflected fields). As other obvious difference is the aniso-tropy of the device, which is only able to reproduce two-dimensional sources, of(practically) infinite extent in the z-direction (i.e., the direction of magnetization).The main advantage of near-field subdiffraction imaging by ferrite slabs is tunability.In fact, the frequency of operation of such devices can be tuned by changing the valueof the external magnetization H0. This can be a crucial advantage, because all sub-diffraction imaging devices reported to date are intrinsically very narrow-banddevices. Quasistatic subdiffraction imaging by ferrite slabs was first reported in[31]. In this paper, the relation between the whole fields at the image and sourceplanes was analyzed. It was shown that the field at the image plane is a combinationof the field at the source plane and its Hilbert transform.

5.4.4 Imaging by Resonant Impedance Surfaces:Magneto-Inductive Lenses

Let us consider an arbitrary surface, characterized by some transverse transmissionmatrix

Eþ1

E�1

� �; T

¼1 �

Eþ2

E�2

� �¼ a b

c d

� �� Eþ

2E�2

� �: (5:87)

The corresponding transverse transmission matrix for a pair of identical coupledsurfaces, separated by a distance d (see Fig. 5.16) will be

T¼2 ¼

a bc d

� �� ead 0

0 e�ad

� �� a b

c d

� �(5:88)

or

T¼2 ¼ a2 ead þ bc e�ad ab ead þ bd e�ad

ac ead þ c2 e�ad cb ead þ d2 e�ad

� �, (5:89)

where only evanescent waves in the transverse direction are considered, and a(v) isthe attenuation constant of such waves.17 It is clear that, in order to have the propertransmission coefficient for evanescent wave amplification through the device,(t ¼ 1=T11 / exp{ad}) it must be that

a ¼ 0 and bc ¼ constant: (5:90)

17In the quasistatic limit, where the effects of the displacement current are neglected, a ¼ jkj, where k is thepropagation constant of the considered surface wave.


It can be easily realized that the first condition corresponds to the excitation of surfacewaves in a single impedance surface (allowing for solutions to equation 5.87with Eþ

1 ¼ E�2 ¼ 0). However, it must be taken into account that, in order to have

an imaging device, the above condition must be simultaneously satisfied for allvalues of the propagation constant k. In practice, this means that the dispersionrelation for such surface waves must be very flat, so that all the wavevectors areexcited in a small frequency band. If this condition is fulfilled, the goal is to obtain

a(v, k) � 0 and b(v, k)c(v, k) � constant, (5:91)

for a wide region of the (v, k) plane.As has already been mentioned, the key concepts of subdiffraction imaging

devices made of a pair of coupled arrays of resonators were first proposed in [22].Subsequently, this approach has been applied in [45] to the analysis of themagneto-inductive lens previously reported in [27]. As was shown in Section5.3.2, a two-dimensional array of magnetic resonators—such as SRRs or capacitivelyloaded metallic rings—can support MIWs whose dispersion relation is given (in thenearest-neighbor approximation) by equation (5.43). Such surface MIWs can beexcited in the interval jv� v0j dv, where

dv ¼ 2v0M

L: (5:92)

Except for very close rings, M=L . 0:1, which implies that a pair of coupledmagneto-inductive surfaces can be a good candidate for the design of subdiffractionimaging devices. The transfer function for such devices is the transmission coefficient

FIGURE 5.16 Two coupled impedance surfaces, separated by a distance d.


FIGURE 5.17 (a) Three-dimensional plot of the transfer function T(kx, ky ¼ 0, v) for amagneto-inductive lens such as that shown in Figure 5.16 when the surfaces at x ¼ x1, x2are square arrays of capacitively coupled rings. The transfer function is calculated for the fol-lowing parameters: distance d ¼ 4.5mm, periodicity a ¼ 12.5mm, ring resistance R ¼ 0.04V,ring capacitance C ¼ 82pF, ring inductance L ¼ 16.3nH, frequency of resonance of the ringsv0 ¼ 2p� 137:5 MHz, and ring inductanceM ¼ 20.015L. The average radius of the rings isr ¼ 4.5mm. (b) Two-dimensional plot of the transfer function T(kx, ky ¼ 0, v) atv ¼ 2p� 140 MHz. (Source: Data reproduced with permission from [45], copyright 2006,American Institute of Physics.)


between x ¼ 0 and x ¼ 2d in Figure 5.16

T(jkj, v) ¼ e�ad

[T2]11¼ e�jkjd

a2 ejkjd þ bc e�jkjd ; jkj ¼ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y

q, (5:93)

where, from equation (5.56)

a bc d

� �¼ 1

22þ Z0=Za Z0=Za�Z0=Za 2� Z0=Za

� �, (5:94)

with Z0 and Za given by equations (5.48) and (5.54), respectively. Ifa ¼ 2þ Z0=Za � 0, then bc � �1, and equation (5.91) is satisfied. Figure 5.17shows a three-dimensional plot of the transfer function (5.93) for two coupledsquare arrays of capacitively coupled rings of periodicity a, separated a distanced. The plot is shown in the first Brillouin zone 0 , k , p=a. As can be seen, aflat transfer function with jT(jkj, v)j � 1 is obtained at frequencies around thefrequency of resonance of the rings, which is also the center of the bandwidth forthe excitation of MIWs in a single array. Such a flat transfer function frequencyregion is limited by two crests, which corresponds to the two MIWs that can beexcited in the pair of coupled magneto-inductive arrays. As has already beenmentioned, this behavior is typical of a subdiffraction imaging device. It can alsobe seen from equations (5.89) and (5.90) that the reported magneto-inductivelens (as well as any other similar lens made from two identical resonant impedancesurfaces) is not matched to free space. That is, it produces reflected fields that canaffect the source field. However, by using two different surfaces, it is possible—atleast theoretically—to obtain a matched lens [22] (see Problem 5.10). An extensiveanalysis of magneto-inductive lenses, including experiments, has been reportedin [45].

5.4.5 Canalization Devices

Canalization devices are devices that are able to translate a field distribution, pixel topixel, from one interface to another. There can be many designs allowing suchimage translation, and probably the simpler one is the wire media lens proposedin [46] and subsequently demonstrated in [34]. The device is sketched inFigure 5.18, and is made from a square array of conducting wires of periodicitya � l and length l ¼ nl=2, where l is the free-space wavelength at the operationfrequency. The device can reproduce at the exit interface any quasielectrostaticfield distribution imposed at the input interface, with a resolution that is limitedby the array periodicity a. This behavior has been explained as a consequence ofthe specific properties of the homogenized wire media [47]. In this paragraph wewill give an alternative and perhaps even simpler explanation. The wire array canbe seen as a multiconductor TEM transmission line that supports N 2 1 orthogonal


degenerate TEM modes (N is the total number of wires), all of them propagatingwith the vacuum phase constant k ¼ v


p.18 If the condition l ¼ nl=2 is

imposed on the total length of the device, it becomes a Fabry–Perot resonator,which can support standing waves corresponding to any combination of the afore-mentioned TEM modes, bouncing back and forth between both ends of thedevice. These resonator modes, which can have an arbitrary voltage distributionon the wires, have zero currents and maximum voltages on each wire end. Whena given quasielectrostatic voltage distribution is imposed at the input interface(l ¼ 0), it excites the resonator mode that most closely matches this voltage distri-bution, which is reproduced at the output interface (l ¼ nl=2) with the aforemen-tioned resolution a. In fact, it can be easily realized that the transfer function ofthe device is T ¼ +1, where the + sign applies to even/odd values of n.

The wire lens described above operates in the quasielectrostatic limit; that is, it cancanalize only TM field distributions, being almost insensitive to TE incident fields.One quasimagnetostatic canalization device, able to reproduce TE incident fields,is the swiss-roll hexagonal array proposed in [12] for operation at radio frequencies.19

Canalization of light seems also to be the main mechanism for imaging in somerecently reported photonic crystal devices [35,46].

FIGURE 5.18 Sketch of the wire lens proposed in [46]. (Source: Reprinted with permissionfrom [46], copyright 2005, American Physical Society.)

18It is clear that the simpler choice for this set of orthogonal modes is to define the ith mode by Vi / e+jkz

on the ith wire and V ¼ 0 on the remaining ones.19The mechanism of operation of such device has been further analyzed in [26], where the effect ofmagneto-inductive waves excited in the array was also studied.


PROBLEMS

5.1. Polarizabilities of the CSRR. Using the Babinet principle (5.18), deduceequations (5.6)–(5.8) for a CSRR etched on a perfect conducting plate.

5.2. Circular polarization in SRR arrays. Consider a plane wave incident upona loss-less SRR array. The incident wavevector is in the y–z plane, formingan angle u with the normal to the surface (i.e., with the z-axis). The incidentwave is polarized with the magnetic field perpendicular to the plane of inci-dence (p-polarization). Find, as a function of the SRR polarizabilities, theangle of incidence for which the transmitted wave is circularly polarized.

5.3. Circular polarization in CSRR arrays. Consider the plane wave incidentupon a loss-less CSRR array. Discuss, as a function of the CSRR polarizabil-ities, the angle of incidence and polarization for which the transmitted wave iscircularly polarized.

5.4. Lossy magnetoinductive waves. By including the ohmic resistance of therings in the equation for the current

jvLþ 1jvC

þ R

� �In ¼ M In�1 þ Inþ1ð Þ, (5:95)

find the attenuation constant of MIWs propagating on one-, two-, and three-dimensional arrays of NB-SRRs. Assume that the nearest-neighbor approxi-mation is valid.

5.5. MIWs in sparse arrays of SRRs. By using the dipole approximation (eachSRR is assumed to be equivalent to an elemental dipole), find closedexpressions for the dispersion relation of a magneto-inductive waveguideformed by a one-dimensional coplanar array of capacitively loaded rings. Use

L ¼ m0r ln16rd

� �� 2

� �, (5:96)

where r and d are the ring radius and the wire diameter, respectively, as anapproximation of the ring inductance. Discuss the conditions for the validityof this approach.

5.6. Duality between MIWs/EIWs in NB-SRR/C-NB-SRRs arrays. Takingduality into account, find the relations between the capacitance and the induc-tance of the equivalent circuit of an NB-SRR (see Chapter 2) and those of theequivalent circuit of its complementary, the C-NB-SRR (see Chapter 3). Fromthis result show that, in the limit jMj=L � 1, the dispersion equation (5.57) for


the C-NB-SRR chain of Figure 5.7a is approximately given by

v2

v20

� 1� 2ML

cos (ka), (5:97)

where L and M are the self- and mutual-inductances of the dual chain of NB-SRRs. It is clear that, in the limit jMj=L � 1, this expression coincides with(5.37). This limit is a natural condition for the validity of the assumednearest-neighbor-approach.

5.7. EIWs in two-dimensional CSRR arrays. Find, in terms of the NB-SRRpolarizabilities (see Chapter 1) the dispersion relation for EIWs propagatingon a two-dimensional sparse square array of C-NB-SRRs of periodicity a.

5.8. Surface plasmons in a slab of negative permittivity. Find, from equation(5.67), the dispersion relation k(1; d) for the quasistatic surface plasmons sup-ported by a metallic slab of width d. Assume a Drude–Lorentz model for thepermittivity of the metal.

5.9. MSSWs in a ferrite slab. Find from equation (5.81) the dispersion equationfor MSSWs in a ferrite slab with an external magnetization parallel to the slab.

5.10. Matched impedance-surface lens. Consider two distinct parallel coupledsurface impedances separated by a distance d, as in Figure 5.16. Find the con-ditions that must satisfy its respective array impedances Za and Z0a in order toobtain a perfectly matched lens (t / ejkjd; r ¼ 0) from such configuration(see [22]).

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20. R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar “A theory of metamaterialsbased on periodically loaded transmission lines: Interaction between magnetoinductiveand electromagnetic waves.” J. Appl. Phys., vol. 97, paper 064909, 2005.

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24. M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar “Dispersion characteristicsof magneto-inductive waves: comparison between theory and experiment.” Electron. Lett.,vol. 39, pp. 215–217, 2003.

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REFERENCES 307

INDEX

Admittance surfaces, 268–278All angle negative refraction, 8Alternate right-handed/left-handed

(ARLH) sections, 199–207bandpass filters and, 203–207complementary split ring resonators,

203–207diplexers and, 203–207S-band filter, 202split ring resonators, 199–203

Amplification of evanescent modes, 20–21,288–292

Antenna applications, 252–258angle of main radiation lobe, 254electronically scanned antennas,

254–258leaky-wave antenna, 254–258left-handed media, 254–258media pairs, 253radiation limit of Chu for Q, 254radiated power, 254

Anti-symmetric resonances, 71ARLH. See alternate right-handed/

left-handed.

Babinet principle, 268–269, 272–273complementary split ring resonator

planar arrays, 272–273Backward leaky modes, 19–20, 254–259

Backward media, 2, 3Backward transmission line, 120–128backward wave, 120–121characteristic impedance, 120constitutive parameters, 123–124design and fabrication, 128dispersion of, 123forward wave, 121implementation of, 124–125,

128–131composite right-/left-handed (CRLH),

124–128left-handed structure, 120–121right-handed structure, 121transverse electromagnetic (TEM) mode

propagation, 123–124Backward wave, 120–121Backward wave coupler, 246–248Backward-wave propagation, 2–4, 9–12Cerenkov radiation, 10–11inverse Doppler effect, 10negative Goos–Hanchen shift, 12

Balanced composite right-/left-handed(CRLH) transmission lines, 127–128

Bandpass filters, 198–227admittance inverters, 208alternate right-handed/left-handed

(ARLH) sections, 199–207design methodology, 210


309

Bandpass filters (Continued)hybrid approach, 210hybrid model, 210right-handed section implementation,

218–225ultra-wide, 219–227

Bandwidth, enhancement of, 236–244phase shifters, 244rat race hybrid couplers, 239–243

BC-SRR. See broadside-coupled split-ringresonator design.

Bianisotropy, 65–69Bloch impedance, 120, 152, 155Bloch impedance. See Characteristic

impedanceBroadband device components,

236–244Broadside-coupled split-ring resonator

design (BC-SRR), 60–62Bulk metamaterialsalternative designs, 102–107chiral, 97–102ferrite, 92–97infrared frequencies, at, 79–80optical frequencies, at, 79–80,

106–107split ring resonators (SRRs) based,

65–70, 80–88

Canalization devices, 302–303Cerenkov radiation, 10–11Characteristic (Bloch) impedance, 120Chiral media, 97–102backward-wave propagation, 98–99chiral nihility, 102racemic mixture, 97

Chiral metamaterials, 97–102Chiral nihility, 102Circuit analysis approach, 44Circuit model comparisons, 175–180dual left-handed lines vs. resonant types,

175–180Coefficients, transmission and reflection,

13–15, 17Compact broadband device components,

236–244Complementary split ring resonators

(CSRR), 119, 155–163admittance surfaces, 268–278

alternate right-handed/left-handed(ARLH) sections, 203–207

electromagnetic properties of, 156–160electro-inductive waves, 284–287equivalent circuit models, 156–160,

166–170filters, characteristics of, 207–224frequency-selective surface, 278left-handed transmission lines, 163–166metamaterial transmission line synthesis

and, 163–175negative permittivity, 163–166numerical calculation, 160–163parameter extraction technique, 170–172planar arrays, 272–273

Complex waves, 19–20Composite right-/left-handed (CRLH)

transmission lines, 124–128balanced, 127–128coplanar waveguide (CPW)

configuration, 129host propagating medium, 129

distributed components, 129lumped elements, 129

leaky-wave antennas, 254–255microstrip structure, 129

Constitutive parametersbackward transmission line and,

123–124bulk split-ring metamaterials, 65–70

Coplanar waveguide (CPW), 129, 136Couplers in planar technology, 246–252backward wave, 246–248forward wave, 248–249improvements to, 249–252

CPW. See Coplanar waveguideCRLH. See Composite right-/left-handed

transmission linesCross-polarization effects, 54–59, 99–100,

276–278CSRR. See Complementary split ring

resonators

Diplexers, 188–189alternate right-handed/left-handed

(ARLH) sections, 206–207Dispersion relationsof bulk metamaterials, 69, 83–87,

90–92

310 INDEX

of transmission-line metamaterials,120–128

of resonant type transmission-linemetamaterials, 151, 167

Distributed components, 129Double-negative media, 2Double-split split-ring resonator

(2-SRR), 62Dual left-handed lines, resonant types vs.,

175–180Dual transmission line. See backward

transmission line.Dual transmission line. See purely left-

handed transmission line.Dual-band components, 244–246Duality, 155, 272–273, 285Dynamic resonances, 71

EBGs. See Electromagnetic band gapsEC-SRR. See Edge-coupled split ring

resonatorEdge-coupled split ring resonator

(EC-SRR), 52–59EIWs. See Electro-inductive wavesElectric resonances, 71Electro-inductive wavesapplications of, 287complementary split ring resonators,

284–287Electrodynamics of left-handed media,

1–35Electromagnetic band gap (EBG),

194, 227transmission line, 191–192

Energy density, 4–6Equivalent circuit modelsof composite right/left handed (CRLH)

transmission lines, 124–125of complementary split ring resonators

(CSRRs), 156–160of complementary split ring resonators

(CSRRs) loaded transmissionlines, 166

of purely right/left handed (PRH andPLH) transmission lines, 121

of split ring resonators (SRRs), 52–65of split ring resonators (SRRs) loaded

transmission lines, 146

Evanescent Fourier harmonics, 25–27,289–290

Evanescent mode amplification. SeeAmplification of evanescent modes

Fermat principle, 9Ferrite lens, 296Ferrite metamaterials, 92–97left-handed circularly polarized (LCP)

wave, 93magnetostatic surface

waves (MSSWs), 95right-handed circularly polarized (RCP)

wave, 93Ferrites, low-loss cubic, 92Filters, 188–189complementary split ring resonators

(CSRR), 207–224high-pass, 225–227narrow bandpass, 198–207planar, 193–198S-band, 203tunable, 227–233ultra-wide bandpass, 225–227

Forward leaky modes, 254–259Forward transmission line, phase velocity

calculation, 122Forward wave coupler, 248–249Forward wave, 121Fourier harmonics, 26–27Frequency-selective surface, 278

Goos–Hanchen shift, negative, 12Group velocity, 4–6calculation of, 122

Guided waves, 17–19

Higher-order resonances, 70–73High-pass filters (HPFs), 225–227HPFs. See High-pass filters

In vacuo capacitance, 57, 100Indefinite media, 34–35Infrared frequencies, SRRs at, 75–80Inverse Doppler effect, 10Isotropic split-ring resonators,

73–75

INDEX 311

LCP. See Left-handed circularly polarizedwave

Leaky backward waves, 19–20Left-handed circularly polarized (LCP)

wave, 93Left-handed mediaantenna applications and, 252–258energy density, 4–6Fermat principle, 9group velocity, 4–6impedance, 9, 14, 17, 20, 23, 29impedance matrix, 30left-handed

slabs, 16–20triplet, 3

losses and dispersions, 32–34negative refraction, 6–9other effects, 9–12Poynting vector, 3–6, 36, 41slabs, 1/10! 21 and m/m! 21,

20–32wave fronts, 3–5, 10–11wave numbers, 4, 19, 34wave propagation, 2–4wave vector, 6, 8, 12–15, 19, 35

Left-handed metamaterialschiral bulk, 97–102ferrite bulk, 92–97split ring resonators (SRRs) bulk, 65–70,

80–88planar transmission-line, 120–180resonant type planar transmission line,

135–175resonant/nonresonant planar

transmission line (comparison).175–180

Left-handed slabsguided waves, 17–19leaky waves, 19–20reflection coefficients, 17transmission coefficients, 17with 1/10 !21 and m/m0 !21,

20–32Left-handed transmission lines, equivalent

circuit models, 146–155Left-handed transmission line design, split

ring resonators, 135–146coplanar waveguide, 136CPW technology, 139–143

microstrip line, 136,139–143negative permeability transmission lines,

136–138resonant-type approach, 135size reduction, 144–146

Left-handed triplet, 3Left-handed wavesbackward leaky modes, 19–20complex waves, 19–20guided waves, 17–19surface waves, 15–16, 18TEM waves, 3

Longitudinal section electric (LSE), 13Longitudinal section magnetic

(LSM) waves. See P-polarized wavesLosses and dispersions, 32–34Pendry’s perfect lens, 32super-lens, 33–34Veselago analysis, 32

Low-loss cubic ferrites, 92yttrium iron garnets, 92

Low-loss plasmas, 1LSE. See Longitudinal section electricLSM waves. See P-polarized waves

Magnetic plasma, 105Magnetic resonances, 71Magneto-inductive lenses, 299–302Magneto-inductive waves (MIWs),

278–284applications of, 285–287equation, 279–281surfaces, 282–284

Magnetostatic surface waves (MSSWs), 95.295–299

Matching device, perfect lensand, 29–32

MB. See Mono-bandMedia pairs, 253Metallic plates, 44–46two-dimensional plasmas, 44–46

Metallic waveguides, 44–46magnetic plasma, 105one-dimensional plasmas, 44–46

Metamaterials, left-handed, 80–91Metamaterial transmission lines, balanced

composite right-/left-handed,225–227

Metamaterial transmission line synthesis

312 INDEX

compact broadband devices,236–244

complementary split ring resonators,163–175

equivalent circuit models,166–170

frequency response, 172–175left-handed transmission lines,

163–166negative permittivity, 163–166parameter extraction, 170–172

coupled-line couplers, 246–252dual-band components, 244–246microwave component miniaturization,

234–236Microstrip, 139–143Microwave applications, 187–258Microwave filters, 188–233MIWs. See Magneto inductive wavesMono-band (MB) circuits, 244MSSWs. See Magnetostatic

surface wavesMulti-tuning, stop-band filters and,

190–191

Narrow bandpass filter, 198–207NB-SRR. See Nonbianistropic split ring

resonatorNegative Goos–Hanchen shift, 12Negative magnetic permeabilitybulk split-ring resonator metamaterials,

65–70edge-coupled SSR, 52–59split-ring resonator designs,

59–65synthesis of, 51–80

Negative permeability transmission lines,136–138

Negative permittivity transmission lines,163–166

Negative refraction, 6–9all-angle, 8

Negative 1 and m. See Left-handedNegative-permittivity, 44–50spatial dispersion, 49–50

Negative refractive media. See Left-handedmedia

Nonbianistropic split-ring resonator(NB-SRR), 62

Non-resonant circuit analysisapproach, 44

Notch tunable filters, 230–233

One-dimensional plasmasmetallic waveguides and, 44–46

One-dimensional split-ring resonator basedleft-handed metamaterials, 81–85

Optical frequencies, 75–80, 106–107Open EC-SRR, 111

Parameter extraction technique, 170–172Pendry’s perfect lens, 27–29, 32Perfect lens, 25–32evanescent Fourier harmonics, 26–27tunneling/matching device, 29–32

Perfect tunneling, 21–25Phase compensation, 20–21Phase shifters, 244Phase velocity, calculation of, 122Planar arrayscomplementary split ring resonator,

272–273split ring resonator, 270–272

Planar filters, 188–233Planar technology metamaterials,

119–180backward transmission line, 120–128circuit model comparison, 175–180complementary split rings resonator

(CSRR), 119left-handed transmission lines,

135–146three dimensional metamaterials, 132two-dimensional (2D), 131–134

Plates, metallic, two dimensional plasmas,44–46

PLH. See Purely left-handed transmissionline

P-polarized waves. See Longitudinalsection magnetic (LSM) or transversemagnetic (TM) waves, 13

PRH. See Purely right-handed transmissionline

Purely left-handed (PLH) transmission line,125, 246

dual, 125Purely right-handed (PRH) transmission

line, 125, 246

INDEX 313

Quasielectrostatic limit imaging, 292–295surface plasmons, 292–295

Quasimagnetostatic limit imaging,295–299

magnetostatic surface waves,295–299

Quasistatic resonance, 70

Racemic mixture, 97, 99Radiated power, 254Rat race hybrid couplers, 239–243RCP. See Right-handed circularly polarized

waveReflection coefficients, 17transmission coefficients and, 13–15

Resonances, higher order, 70–73Resonant impedance surface imaging,

299–302magneto-inductive lenses, 299–302

Resonant-type approach, 135Resonant-type balanced composite

right/left-handed metamaterialtransmission lines, 225–227

Resonant-type transmission lines, dualleft-handed vs., 175–180

Right handed sections, bandpass filters and,218–225

Right-handed circularly polarized (RCP)wave, 93

Scaling plasmas at microwave frequencies,44–50

metallic plates, 44–46metallic waveguides, 44–46wire media, 47–50

Silver lens, 294Simultaneously negative 1 and m. See

Left-handedSingle split ring resonators, 268–269Babinet principle, 268–269

Size reduction, split ring resonators and,144–146

Slabs, 1/10 ! 21 and m/m0 ! 21,19–32

perfect lens, 25–32perfect tunneling, 21–25

Spatial dispersion, wire media and, 49–50Spirals, 62–65two-turns spiral resonator (2-SR), 62

Split-ring resonator based left-handedmetamaterials, 80–91

continuous-medium approach, 87–88modeling and numerical accuracy of,

90–91one-dimensional, 81–84superposition hypothesis, 88–90three-dimensional, 85–87two-dimensional, 85–87

Split-ring resonator metamaterials, bulk,65–70

Split-ring resonator designs, 59–65broadside-coupled, 60–62chiral, 99–100double-split, 62nonbianistropic, 62spirals, 62–65

Split-ring resonator planar arrays,270–272

Split-ring resonators (SRRs), 43admittance surfaces, 268–278alternate right-handed/left-handed

(ARLH) sections and, 199–203Babinet principle for, 268–269complementary, 155–163duality, 155–163edge-coupled, 52–59equivalent circuit models, 52–65,

146–155higher-order resonances, 70–73isotropic, 73–75left-handed transmission line design and,

135–146negative permeability transmission lines,

136–138resonant-type approach, 135scaling down of, 75–80

infrared frequencies, 75–80optical frequencies, 75–80

size reduction, 144–146S-polarized waves, 13. See Longitudinal

section electric (LSE); Transverseelectric (TE)

Spurious frequency bands, 193–197SRRs. See Split ring resonatorsStop-band filters, 189–193electromagnetic band gap (EBG)

transmission line, 191–192multi-tuning, 190–191

314 INDEX

Subdiffraction imaging devices, 287–303canalization devices, 302–303features of, 288–292ferrite lens, 296magnetostatic waves, 295–297resonant impedance surfaces, 299–302silver lens, 294surface plasmons, 292–295

Super-lens, 33–34Veselago, 34

Superposition hypothesis, 88–90Surface plasmons, 16, 292–295Surface waves, 15–16, 18magnetostatic, 295–299s-polarized, 16

Symmetric resonances, 71Synthesis of negative magnetic

permeability, 51–80Synthesis, bulk metamaterials, 43–109

TEM. See Transverse electromagnetic modepropagation

Three-dimensional SRR-based left-handedmetamaterials, 85–87

TM waves. See P-polarized wavesTransmission coefficients, 17Transmission, reflection coefficients and,

13–15Transmission lines, negative permeability,

136–138Transverse electric (TE) waves, 13Transverse electromagnetic (TEM) mode

propagation, 123–124Transverse magnetic (TM) waves. See

P-polarized wavesTunable filters, 227–233notch type, 230–233varactor-loaded split rings resonators

(VLSRRs), 227–233Tunnelingperfect, 21–25perfect lens and, 29–32

2-SR. See Two-turns spiral resonator2-SRR. See Double-split split

ring resonatorTwo dimensional left handed structure,

lumped elements and, 131–132

Two dimensional metamaterialsbulk SRR based, 85–87planar technology, 131–135

Two dimensional plasmametallic plates and, 44–46

Two-turns spiral resonator (2-SR), 62

Ultra-wide bandpass filter (UWBPF),219–222, 225–227

resonant-type balanced compositeright/left-handed metamaterialtransmission lines, 225–227

UWBPF. See Ultra-wide bandpass filters

Varactor-loaded split rings resonators(VLSRRs), 227–233

equivalent circuit model, 228–230model validation, 230topology of, 228–230

Veselagoanalysis, 32lens, 34media, 2

VLSRRs. See Varactor-loaded split ringsresonators

Wave impedance, 14Wave propagation, 2–4Wave transmission and guidance,

left-handed slabs and, 16–20Waveguides, metallicas one dimensional magnetic

plasmas, 105as one dimensional plasmas,

44–46Waves at interfaces, 13–16surface, 15–16transmission and reflection coefficients,

13–15Wire media, 47–50spatial dispersion in, 49–50

YIG. See Yttrium iron garnetsYttrium iron garnets (YIG), 92

INDEX 315

metamaterials with

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